One-Sample t Interval for a Population Mean
Estimate a population mean when the population standard deviation is unknown, using the sample standard deviation, a t critical value, and conditions that support the procedure.
One-Sample t Interval for a Population Mean: direct answer
In a one sample t interval problem, begin by identifying the population target and the inferential role of the data. Estimate a population mean when the population standard deviation is unknown, using the sample standard deviation, a t critical value, and conditions that support the procedure.
This page uses one sample t interval as its single primary search focus. The lesson, numerical cases, and retained questions are restricted to that intent so the page does not function as a generic inference question bank.
Quick reference for One-Sample t Interval for a Population Mean
| Parameter | μ: one population mean |
|---|---|
| Point estimate | x̄ |
| Standard error | s/√n |
| Degrees of freedom | n−1 |
| Interval | x̄ ± t* × s/√n |
| Interpretation target | The population mean μ |
Concept mastery: one sample t interval
Use t because sigma is unknown
A one-sample t interval estimates a population mean when the population standard deviation sigma is unknown and the sample standard deviation s is used instead. That extra uncertainty is represented by the t distribution rather than the standard normal distribution.
Center the interval at x-bar
The point estimate is the sample mean x-bar. The interval has the form x-bar ± t-star*s/sqrt(n). The center is determined entirely by the observed sample; the confidence level and sample variability determine how far the endpoints extend from it.
Use degrees of freedom n minus 1
For the standard one-sample t procedure, df=n-1. The t critical value depends on both confidence level and degrees of freedom. As n grows, the t distribution approaches the normal distribution and t-star moves closer to the corresponding z-star.
Check the data-production condition
A random sample or randomized process supports the inferential model. If observations are sampled without replacement from a finite population, the sample-size-to-population relationship matters for approximate independence.
Evaluate shape and outliers, not a ritual phrase
t procedures are robust to moderate nonnormality when the sample is reasonably large, but strong skewness and extreme outliers can be important for small samples. Discuss the actual graph or description supplied in the problem rather than automatically writing “normal condition met.”
Distinguish standard deviation from standard error
s describes variability among individual observations. s/sqrt(n) describes the estimated variability of x-bar across repeated samples. Using s directly in the margin of error makes the interval far too wide.
Interpret the interval for the population mean
The endpoints estimate the population mean mu. They do not contain a fixed percentage of individual observations, and they do not imply that the sample mean has a probability of moving after the data have been collected.
Understand how n changes precision
Increasing sample size shrinks s/sqrt(n) and also slightly reduces the t critical value because degrees of freedom increase. Both effects generally narrow the interval when the sample standard deviation is comparable.
Do not switch to z because n is large
A large sample improves robustness, but it does not make an unknown population standard deviation suddenly known. In standard AP Statistics inference, one-sample mean intervals use t when sigma is unknown.
Tie the conclusion to the measured variable and population
A complete interpretation names the population, the quantitative variable, the confidence level, and the interval endpoints with units. This keeps the statistical object connected to the question that generated the data.
Worked analysis for one sample t interval
Mean interval case 1: State Park
A random sample of n=18 has x̄=42.00 minutes and s=5.50 minutes. Because the population standard deviation is unknown, the one sample t interval uses df=17, SE=s/√n=1.296, and t*=1.740 for 90% confidence. The margin of error is 2.255 minutes, producing (39.74, 44.26) minutes.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=18; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 2: Food Cooperative
A random sample of n=21 has x̄=44.70 points and s=6.30 points. Because the population standard deviation is unknown, the one sample t interval uses df=20, SE=s/√n=1.375, and t*=2.086 for 95% confidence. The margin of error is 2.868 points, producing (41.83, 47.57) points.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=21; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 3: University Advising Center
A random sample of n=24 has x̄=47.40 hours and s=7.10 hours. Because the population standard deviation is unknown, the one sample t interval uses df=23, SE=s/√n=1.449, and t*=2.807 for 99% confidence. The margin of error is 4.069 hours, producing (43.33, 51.47) hours.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=24; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 4: Regional Manufacturer
A random sample of n=27 has x̄=50.10 dollars and s=7.90 dollars. Because the population standard deviation is unknown, the one sample t interval uses df=26, SE=s/√n=1.520, and t*=1.706 for 90% confidence. The margin of error is 2.593 dollars, producing (47.51, 52.69) dollars.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=27; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 5: Public Health Clinic
A random sample of n=30 has x̄=52.80 grams and s=8.70 grams. Because the population standard deviation is unknown, the one sample t interval uses df=29, SE=s/√n=1.588, and t*=2.045 for 95% confidence. The margin of error is 3.249 grams, producing (49.55, 56.05) grams.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=30; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 6: Urban Recreation Program
A random sample of n=33 has x̄=55.50 days and s=9.50 days. Because the population standard deviation is unknown, the one sample t interval uses df=32, SE=s/√n=1.654, and t*=2.738 for 99% confidence. The margin of error is 4.529 days, producing (50.97, 60.03) days.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=33; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 7: School District
A random sample of n=36 has x̄=58.20 minutes and s=10.30 minutes. Because the population standard deviation is unknown, the one sample t interval uses df=35, SE=s/√n=1.717, and t*=1.690 for 90% confidence. The margin of error is 2.900 minutes, producing (55.30, 61.10) minutes.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=36; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 8: Local Election Office
A random sample of n=39 has x̄=60.90 points and s=11.10 points. Because the population standard deviation is unknown, the one sample t interval uses df=38, SE=s/√n=1.777, and t*=2.024 for 95% confidence. The margin of error is 3.598 points, producing (57.30, 64.50) points.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=39; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 9: Energy-Efficiency Pilot
A random sample of n=42 has x̄=63.60 hours and s=11.90 hours. Because the population standard deviation is unknown, the one sample t interval uses df=41, SE=s/√n=1.836, and t*=2.701 for 99% confidence. The margin of error is 4.960 hours, producing (58.64, 68.56) hours.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=42; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 10: Community Broadband Project
A random sample of n=45 has x̄=66.30 dollars and s=5.50 dollars. Because the population standard deviation is unknown, the one sample t interval uses df=44, SE=s/√n=0.820, and t*=1.680 for 90% confidence. The margin of error is 1.378 dollars, producing (64.92, 67.68) dollars.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=45; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 11: Regional Bus Network
A random sample of n=48 has x̄=69.00 grams and s=6.30 grams. Because the population standard deviation is unknown, the one sample t interval uses df=47, SE=s/√n=0.909, and t*=2.012 for 95% confidence. The margin of error is 1.829 grams, producing (67.17, 70.83) grams.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=48; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 12: Campus Dining Service
A random sample of n=51 has x̄=71.70 days and s=7.10 days. Because the population standard deviation is unknown, the one sample t interval uses df=50, SE=s/√n=0.994, and t*=2.678 for 99% confidence. The margin of error is 2.662 days, producing (69.04, 74.36) days.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=51; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value.
Mean interval case 13: Workforce Training Program
A random sample of n=18 has x̄=74.40 minutes and s=7.90 minutes. Because the population standard deviation is unknown, the one sample t interval uses df=17, SE=s/√n=1.862, and t*=1.740 for 90% confidence. The margin of error is 3.239 minutes, producing (71.16, 77.64) minutes.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=18; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 13: Workforce Training Program context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Mean interval case 14: County Recycling Audit
A random sample of n=21 has x̄=77.10 points and s=8.70 points. Because the population standard deviation is unknown, the one sample t interval uses df=20, SE=s/√n=1.898, and t*=2.086 for 95% confidence. The margin of error is 3.960 points, producing (73.14, 81.06) points.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=21; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 14: County Recycling Audit context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Mean interval case 15: University Residence Halls
A random sample of n=24 has x̄=79.80 hours and s=9.50 hours. Because the population standard deviation is unknown, the one sample t interval uses df=23, SE=s/√n=1.939, and t*=2.807 for 99% confidence. The margin of error is 5.444 hours, producing (74.36, 85.24) hours.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=24; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 15: University Residence Halls context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Mean interval case 16: Telehealth Pilot
A random sample of n=27 has x̄=44.50 dollars and s=10.30 dollars. Because the population standard deviation is unknown, the one sample t interval uses df=26, SE=s/√n=1.982, and t*=1.706 for 90% confidence. The margin of error is 3.381 dollars, producing (41.12, 47.88) dollars.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=27; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 16: Telehealth Pilot context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Mean interval case 17: Public Museum
A random sample of n=30 has x̄=47.20 grams and s=11.10 grams. Because the population standard deviation is unknown, the one sample t interval uses df=29, SE=s/√n=2.027, and t*=2.045 for 95% confidence. The margin of error is 4.145 grams, producing (43.06, 51.34) grams.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=30; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 17: Public Museum context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Mean interval case 18: Youth Sports League
A random sample of n=33 has x̄=49.90 days and s=11.90 days. Because the population standard deviation is unknown, the one sample t interval uses df=32, SE=s/√n=2.072, and t*=2.738 for 99% confidence. The margin of error is 5.673 days, producing (44.23, 55.57) days.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=33; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 18: Youth Sports League context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Mean interval case 19: Rural Pharmacy Network
A random sample of n=36 has x̄=52.60 minutes and s=5.50 minutes. Because the population standard deviation is unknown, the one sample t interval uses df=35, SE=s/√n=0.917, and t*=1.690 for 90% confidence. The margin of error is 1.549 minutes, producing (51.05, 54.15) minutes.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=36; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 19: Rural Pharmacy Network context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Mean interval case 20: City Tree Program
A random sample of n=39 has x̄=55.30 points and s=6.30 points. Because the population standard deviation is unknown, the one sample t interval uses df=38, SE=s/√n=1.009, and t*=2.024 for 95% confidence. The margin of error is 2.042 points, producing (53.26, 57.34) points.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=39; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 20: City Tree Program context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Mean interval case 21: District Tutoring Initiative
A random sample of n=42 has x̄=58.00 hours and s=7.10 hours. Because the population standard deviation is unknown, the one sample t interval uses df=41, SE=s/√n=1.096, and t*=2.701 for 99% confidence. The margin of error is 2.959 hours, producing (55.04, 60.96) hours.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=42; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 21: District Tutoring Initiative context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Mean interval case 22: Regional Call Center
A random sample of n=45 has x̄=60.70 dollars and s=7.90 dollars. Because the population standard deviation is unknown, the one sample t interval uses df=44, SE=s/√n=1.178, and t*=1.680 for 90% confidence. The margin of error is 1.979 dollars, producing (58.72, 62.68) dollars.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=45; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 22: Regional Call Center context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Mean interval case 23: City Transit Survey
A random sample of n=48 has x̄=63.40 grams and s=8.70 grams. Because the population standard deviation is unknown, the one sample t interval uses df=47, SE=s/√n=1.256, and t*=2.012 for 95% confidence. The margin of error is 2.526 grams, producing (60.87, 65.93) grams.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=48; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 23: City Transit Survey context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Mean interval case 24: Regional Hospital
A random sample of n=51 has x̄=66.10 days and s=9.50 days. Because the population standard deviation is unknown, the one sample t interval uses df=50, SE=s/√n=1.330, and t*=2.678 for 99% confidence. The margin of error is 3.562 days, producing (62.54, 69.66) days.
The interval estimates the population mean, not the range containing a fixed percentage of individual measurements. The sampling or assignment mechanism must support inference, and shape/outlier information should be evaluated in light of n=51; smaller samples need stronger protection from extreme skewness or influential outliers. Increasing n would generally reduce the standard error, while raising the confidence level would increase the critical value. In the Mean interval case 24: Regional Hospital context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.
Robustness case 25: Mean estimation
For n=14, x̄=68.4, and s=11.8, a 95% one sample t interval uses df=13, SE=3.154, and t*=2.160. The margin of error is 6.813, so the interval is (61.59, 75.21). Because sigma is not known, using a z critical value solely because n=14 would not match the standard t procedure.
The interpretation depends on the shape information that accompanies this sample. At n=14, an extreme outlier or severe skew would deserve careful attention before trusting the interval. The interval estimates the population mean; it is not a prediction interval for individual observations. Its width reflects both the sample variability s=11.8 and the uncertainty reduction from √n.
Robustness case 26: Mean estimation
For n=27, x̄=52.1, and s=7.3, a 90% one sample t interval uses df=26, SE=1.405, and t*=1.706. The margin of error is 2.396, so the interval is (49.70, 54.50). Because sigma is not known, using a z critical value solely because n=27 would not match the standard t procedure.
The interpretation depends on the shape information that accompanies this sample. At n=27, an extreme outlier or severe skew would deserve careful attention before trusting the interval. The interval estimates the population mean; it is not a prediction interval for individual observations. Its width reflects both the sample variability s=7.3 and the uncertainty reduction from √n.
Robustness case 27: Mean estimation
For n=45, x̄=103.6, and s=19.2, a 99% one sample t interval uses df=44, SE=2.862, and t*=2.692. The margin of error is 7.706, so the interval is (95.89, 111.31). Because sigma is not known, using a z critical value solely because n=45 would not match the standard t procedure.
The interpretation depends on the shape information that accompanies this sample. At n=45, the t procedure has more robustness to moderate nonnormality, though influential outliers still matter. The interval estimates the population mean; it is not a prediction interval for individual observations. Its width reflects both the sample variability s=19.2 and the uncertainty reduction from √n.
Robustness case 28: Mean estimation
For n=72, x̄=31.8, and s=6.5, a 95% one sample t interval uses df=71, SE=0.766, and t*=1.994. The margin of error is 1.527, so the interval is (30.27, 33.33). Because sigma is not known, using a z critical value solely because n=72 would not match the standard t procedure.
The interpretation depends on the shape information that accompanies this sample. At n=72, the t procedure has more robustness to moderate nonnormality, though influential outliers still matter. The interval estimates the population mean; it is not a prediction interval for individual observations. Its width reflects both the sample variability s=6.5 and the uncertainty reduction from √n.
One Sample T Interval multiple-choice practice
Question 1. One-Sample t Interval
A random sample of 44 travelers at a regional airport authority in Lakeside district during a follow-up evaluation period has mean 49.8 and SD 8.56 for security wait time. Construct a 90% t interval for the population mean.
Answer: A
df=43, t*=1.681, SE=s√n=8.56√44=1.29, so the interval is 49.8±2.169=(47.63, 51.97). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 2. One-Sample t Interval
A random sample of 29 calls at a municipal emergency dispatch center in Central County during a follow-up evaluation period has mean 49.49 and SD 7.24 for response time. Construct a 90% t interval for the population mean.
Answer: B
df=28, t*=1.701, SE=s√n=7.24√29=1.344, so the interval is 49.49±2.287=(47.2, 51.78). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 3. One-Sample t Interval
A random sample of 18 installations at a solar installer in Lakeside district during a summer implementation review has mean 68.73 and SD 15.95 for daily energy output. Construct a 90% t interval for the population mean.
Answer: D
df=17, t*=1.74, SE=s√n=15.95√18=3.759, so the interval is 68.73±6.54=(62.19, 75.27). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 4. One-Sample t Interval
A random sample of 29 accounts at a municipal water office in Prairie District during a six-week field trial has mean 80.53 and SD 17.65 for monthly household use. Construct a 95% t interval for the population mean.
Answer: B
df=28, t*=2.048, SE=s√n=17.65√29=3.278, so the interval is 80.53±6.714=(73.82, 87.24). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 5. One-Sample t Interval
A random sample of 39 enrolled learners at a community college in Atlantic Corridor during a six-week field trial has mean 77.93 and SD 14.9 for course completion. Construct a 95% t interval for the population mean.
Answer: C
df=38, t*=2.024, SE=s√n=14.9√39=2.386, so the interval is 77.93±4.83=(73.1, 82.76). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 6. One-Sample t Interval
A random sample of 17 samples at a food safety laboratory in Mountain Region during a yearly program evaluation has mean 82.56 and SD 8.58 for sample concentration. Construct a 95% t interval for the population mean.
Answer: B
df=16, t*=2.12, SE=s√n=8.58√17=2.081, so the interval is 82.56±4.411=(78.15, 86.97). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 7. One-Sample t Interval
A random sample of 34 customers at a grocery cooperative in Great Lakes during a weekday operations study has mean 84.26 and SD 8.68 for checkout time. Construct a 90% t interval for the population mean.
Answer: D
df=33, t*=1.692, SE=s√n=8.68√34=1.489, so the interval is 84.26±2.519=(81.74, 86.78). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 8. One-Sample t Interval
A random sample of 40 residents at a public health department in Prairie District during a weekday operations study has mean 40.43 and SD 17.1 for vaccination appointment completion. Construct a 90% t interval for the population mean.
Answer: C
df=39, t*=1.685, SE=s√n=17.1√40=2.704, so the interval is 40.43±4.555=(35.87, 44.99). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 9. One-Sample t Interval
A random sample of 33 students at a public high school in Pacific Northwest during a pre-exam training cycle has mean 59.8 and SD 13.47 for algebra benchmark completion. Construct a 90% t interval for the population mean.
Answer: C
df=32, t*=1.694, SE=s√n=13.47√33=2.345, so the interval is 59.8±3.972=(55.83, 63.77). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 10. One-Sample t Interval
A random sample of 20 animals at a wildlife clinic in Desert County during a spring 2027 pilot has mean 44.09 and SD 16.6 for recovery time. Construct a 90% t interval for the population mean.
Answer: B
df=19, t*=1.729, SE=s√n=16.6√20=3.712, so the interval is 44.09±6.418=(37.67, 50.51). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 11. One-Sample t Interval
A random sample of 45 students at a school district in Riverbend during a six-week field trial has mean 86.25 and SD 6.43 for lunch-program participation. Construct a 90% t interval for the population mean.
Answer: A
df=44, t*=1.68, SE=s√n=6.43√45=0.959, so the interval is 86.25±1.611=(84.64, 87.86). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 12. One-Sample t Interval
A random sample of 31 visitors at a county library in Prairie District during a fall 2026 audit has mean 62.93 and SD 17.07 for weekly program attendance. Construct a 90% t interval for the population mean.
Answer: B
df=30, t*=1.697, SE=s√n=17.07√31=3.066, so the interval is 62.93±5.204=(57.73, 68.13). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 13. One-Sample t Interval
A random sample of 25 travelers at a regional airport authority in Pacific Northwest during a fall 2026 audit has mean 68.93 and SD 15.11 for security wait time. Construct a 95% t interval for the population mean.
Answer: B
df=24, t*=2.064, SE=s√n=15.11√25=3.022, so the interval is 68.93±6.237=(62.69, 75.17). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 14. One-Sample t Interval
A random sample of 29 travelers at a regional airport authority in Metro East during a school-year data collection has mean 35.44 and SD 16.92 for security wait time. Construct a 90% t interval for the population mean.
Answer: A
df=28, t*=1.701, SE=s√n=16.92√29=3.142, so the interval is 35.44±5.345=(30.1, 40.78). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 15. One-Sample t Interval
A random sample of 19 enrolled learners at a community college in Mountain Region during a spring 2027 pilot has mean 67.91 and SD 13.85 for course completion. Construct a 95% t interval for the population mean.
Answer: D
df=18, t*=2.101, SE=s√n=13.85√19=3.177, so the interval is 67.91±6.675=(61.23, 74.59). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 16. One-Sample t Interval
A random sample of 40 customers at a grocery cooperative in Sunbelt district during a service-improvement study has mean 72.44 and SD 7.33 for checkout time. Construct a 90% t interval for the population mean.
Answer: C
df=39, t*=1.685, SE=s√n=7.33√40=1.159, so the interval is 72.44±1.953=(70.49, 74.39). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 17. One-Sample t Interval
A random sample of 37 parts at a regional manufacturer in Desert County during a fall 2026 audit has mean 76.17 and SD 8.38 for part diameter. Construct a 90% t interval for the population mean.
Answer: C
df=36, t*=1.688, SE=s√n=8.38√37=1.378, so the interval is 76.17±2.326=(73.84, 78.5). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 18. One-Sample t Interval
A random sample of 34 animals at a wildlife clinic in North Valley during a randomized pilot period has mean 37.15 and SD 7.91 for recovery time. Construct a 90% t interval for the population mean.
Answer: B
df=33, t*=1.692, SE=s√n=7.91√34=1.357, so the interval is 37.15±2.296=(34.85, 39.45). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 19. One-Sample t Interval
A random sample of 24 appointments at a university advising center in Riverbend during a two-month observation window has mean 73.88 and SD 6.32 for appointment wait time. Construct a 95% t interval for the population mean.
Answer: B
df=23, t*=2.069, SE=s√n=6.32√24=1.29, so the interval is 73.88±2.669=(71.21, 76.55). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Question 20. One-Sample t Interval
A random sample of 42 travelers at a regional airport authority in New England network during a multiweek validation study has mean 69.88 and SD 11.01 for security wait time. Construct a 90% t interval for the population mean.
Answer: C
df=41, t*=1.683, SE=s√n=11.01√42=1.699, so the interval is 69.88±2.859=(67.02, 72.74). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
One Sample T Interval free-response practice
FRQ set 1: One-Sample t Interval
Scenario. A random sample of 26 enrolled learners at a community college in Atlantic Corridor during a fall 2026 audit has mean 71.87 and SD 10.21 for course completion. Construct a 90% t interval for the population mean.
- Define the population parameter and preserve the stated group order.
- Verify randomization, independence, and procedure-specific success/failure or shape conditions.
- Construct the interval from the statistic, critical value, and standard error.
- Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.
Model response
df=25, t*=1.708, SE=s√n=10.21√26=2.002, so the interval is 71.87±3.42=(68.45, 75.29). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
FRQ set 2: One-Sample t Interval
Scenario. A random sample of 29 calls at a municipal emergency dispatch center in North Valley during a spring 2027 pilot has mean 35.37 and SD 15.63 for response time. Construct a 95% t interval for the population mean.
- Define the population parameter and preserve the stated group order.
- Verify randomization, independence, and procedure-specific success/failure or shape conditions.
- Construct the interval from the statistic, critical value, and standard error.
- Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.
Model response
df=28, t*=2.048, SE=s√n=15.63√29=2.902, so the interval is 35.37±5.945=(29.42, 41.32). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
FRQ set 3: One-Sample t Interval
Scenario. A random sample of 40 residents at a public health department in Riverbend during a six-week field trial has mean 45.18 and SD 8.5 for vaccination appointment completion. Construct a 95% t interval for the population mean.
- Define the population parameter and preserve the stated group order.
- Verify randomization, independence, and procedure-specific success/failure or shape conditions.
- Construct the interval from the statistic, critical value, and standard error.
- Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.
Model response
df=39, t*=2.023, SE=s√n=8.5√40=1.344, so the interval is 45.18±2.718=(42.46, 47.9). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
FRQ set 4: One-Sample t Interval
Scenario. A random sample of 33 students at a public high school in South Harbor during a spring 2027 pilot has mean 65.9 and SD 6.47 for algebra benchmark completion. Construct a 90% t interval for the population mean.
- Define the population parameter and preserve the stated group order.
- Verify randomization, independence, and procedure-specific success/failure or shape conditions.
- Construct the interval from the statistic, critical value, and standard error.
- Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.
Model response
df=32, t*=1.694, SE=s√n=6.47√33=1.126, so the interval is 65.9±1.908=(63.99, 67.81). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
FRQ set 5: One-Sample t Interval
Scenario. A random sample of 40 bus trips at a city transit agency in Pine Ridge during a winter readiness review has mean 42.9 and SD 6.51 for on-time arrival. Construct a 95% t interval for the population mean.
- Define the population parameter and preserve the stated group order.
- Verify randomization, independence, and procedure-specific success/failure or shape conditions.
- Construct the interval from the statistic, critical value, and standard error.
- Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.
Model response
df=39, t*=2.023, SE=s√n=6.51√40=1.029, so the interval is 42.9±2.082=(40.82, 44.98). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
FRQ set 6: One-Sample t Interval
Scenario. A random sample of 42 patients at a regional hospital in Metro East during a spring 2027 pilot has mean 76.61 and SD 14.04 for appointment completion. Construct a 90% t interval for the population mean.
- Define the population parameter and preserve the stated group order.
- Verify randomization, independence, and procedure-specific success/failure or shape conditions.
- Construct the interval from the statistic, critical value, and standard error.
- Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.
Model response
df=41, t*=1.683, SE=s√n=14.04√42=2.166, so the interval is 76.61±3.646=(72.96, 80.26). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
FRQ set 7: One-Sample t Interval
Scenario. A random sample of 43 parts at a regional manufacturer in Sunbelt district during a weekday operations study has mean 75.6 and SD 14.99 for part diameter. Construct a 95% t interval for the population mean.
- Define the population parameter and preserve the stated group order.
- Verify randomization, independence, and procedure-specific success/failure or shape conditions.
- Construct the interval from the statistic, critical value, and standard error.
- Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.
Model response
df=42, t*=2.018, SE=s√n=14.99√43=2.286, so the interval is 75.6±4.613=(70.99, 80.21). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
FRQ set 8: One-Sample t Interval
Scenario. A random sample of 37 travelers at a regional airport authority in Westview during a school-year data collection has mean 64.59 and SD 8.52 for security wait time. Construct a 95% t interval for the population mean.
- Define the population parameter and preserve the stated group order.
- Verify randomization, independence, and procedure-specific success/failure or shape conditions.
- Construct the interval from the statistic, critical value, and standard error.
- Interpret the confidence statement for the population parameter and explain one factor affecting margin of error.
Model response
df=36, t*=2.028, SE=s√n=8.52√37=1.401, so the interval is 64.59±2.841=(61.75, 67.43). The interpretation concerns the population mean, conditional on randomization/independence and an acceptable population shape or robust sample size.
Next steps after mastering one sample t interval
After mastering the one-sample t interval, compare it with the one-sample t test and with two-sample or paired t procedures. For each new problem, decide first whether there is one quantitative sample, two independent groups, or matched observations. Procedure selection should come from the data structure before any calculator command is chosen.
For precision practice, hold x̄ and s fixed while changing n and confidence level. Observe that larger n shrinks s/√n and changes degrees of freedom, whereas higher confidence enlarges t*. Then inspect a small-sample example with an outlier to decide whether the numerical interval is trustworthy enough to interpret.
