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Relationship between Hypothesis Testing and Confidence Intervals in Statistical Analysis, Exams of Statistics
The connection between hypothesis testing and confidence intervals using examples. It covers one-sided and two-sided tests, and provides instructions for calculating confidence intervals and interpreting the results. The document also includes practice problems for estimating population parameters and testing hypotheses.
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Stat 4473 – Data Analysis Relationship between hypothesis testing and confidence intervals (Homework on back side)
Suppose the significance level is preset at ". The relationship between hypothesis testing and confidence intervals is described below.
(1) H (^) o: population parameter = hypothesized value H 1 : population parameter … hypothesized value
Reject Ho in favor of H 1 if the hypothesized value is outside the 100(1 - ")% confidence interval (two-sided) for the population parameter
Example: Ho: μ = 5 H 1 : μ… 5
! If a 95% confidence interval for : doesn’t contain 5, then we know the p-value for the test is less than .05.
If a 95% confidence interval for : does contain 5, then we know the p-value is greater than.
! If a 90% confidence interval for : doesn’t contain 5, then we know the p-value for the test is less than .10.
If a 90% confidence interval for : does contain 5, then we know the p-value for the test is greater than .10.
! If a 99% confidence interval for : doesn’t contain 5, then we know the p-value for the test is less than .01.
If a 99% confidence interval for : does contain 5, then we know the p-value for the test is greater than .01.
(2) H (^) o: population parameter = hypothesized value H 1 : population parameter > hypothesized value
Reject Ho in favor of H 1 if the hypothesized value is outside the 100(1 - ")% lower confidence interval (one-sided) for the population parameter.
A 100(1 - ")% lower confidence interval is an interval of the form (left endpt., + 4 ). It says that we are 100(1 - ")% confident that the population parameter is at least as large as the left endpoint. (It gives just a lower limit.) All the margin of error is on the right side.
(3) H (^) o: population parameter = hypothesized value H 1 : population parameter < hypothesized value
Reject Ho in favor of H 1 if the hypothesized value is outside the 100(1 - ")% upper confidence interval (one-sided) for the population parameter.
A 100(1 - ")% upper confidence interval is an interval of the form (!4, right endpt). It says that we are 100(1 - ")% confident that the population parameter is no larger than the right endpoint. (It gives just an upper limit.) All the margin of error is on the left side.
Stat 4473 – Data Analysis Problems Involving the Relationship Between Hypothesis Testing and Confidence Intervals
- The following data are the lengths of ears (in cm) collected in a study of size inheritance in a hybrid variety of corn:
18.1 15.3 14.4 10.5 18.2 18.7 16.4 15.8 17.1 12.4 14. 16.2 11.5 14.7 13.
(a) The researchers are interested in whether the average length of ears for the hybrid variety is different from 17 cm. Perform the appropriate test of hypothesis.
(b) Estimate the average length of ears for the hybrid variety with a 95% confidence interval, and interpret your interval estimate.
(c) Does the 95% confidence interval in part (b) contain 17? Explain why that makes sense, based on the results of your hypothesis test in part (a).
(d) Now, estimate the average length of ears for the hybrid variety with a 99% confidence interval. Does the 99% confidence interval contain 17? Explain why that makes sense, based on the results of your hypothesis test in part (a).
- The following data are measures of pulmonary vascular resistance (PVR) in eight lambs before and after infusion of the drug histamine.
Lamb PVR Before PVR After Histamine Histamine
1 0.095 0. 2 0.106 0. 3 0.082 0. 4 0.152 0. 5 0.090 0. 6 0.086 0. 7 0.137 0. 8 0.121 0.
(a) Perform a statistical test to determine whether histamine increases PVR on average. Use a preset significance level of " = .10.
(b) Estimate the mean difference in the PVR levels with a 90% confidence interval. Explain why the interpretation of your confidence interval may not be consistent with the results from part (a).
(c) Now construct the appropriate 90% one-sided confidence interval for the mean difference in PVR levels. Interpret your interval estimate.