1. A certain population follows a normal distribution, with mean m and standard deviation s = 2.5. You collect data and test the hypotheses
H0: m = 1, Ha: m 1.
You obtain a P-value of 0.022. Which of the following is true? Why?
a. A 95% confidence interval for m will include the value 1.
b. A 95% confidence interval for m will include the value 0.
c. A 99% confidence interval for m will include the value 1.
d. A 99% confidence interval for m will include the value 0.
To determine which statement is true, we need to understand the concept of the confidence interval and its relationship with hypothesis testing.
A confidence interval provides a range of values within which the true population parameter is likely to fall with a certain level of confidence. In this case, we are interested in the population mean, denoted by "m."
Hypothesis testing involves comparing a null hypothesis (H0) with an alternative hypothesis (Ha) and evaluating the evidence from the sample to determine whether to accept or reject the null hypothesis. The P-value, in this case, represents the probability of observing a sample mean as extreme as or more extreme than the one obtained if the null hypothesis is true.
Now let's evaluate the given options:
a. A 95% confidence interval for m will include the value 1.
This statement is not necessarily true based solely on the P-value. A confidence interval is unrelated to the P-value in hypothesis testing. The P-value only provides a measure of evidence against the null hypothesis.
b. A 95% confidence interval for m will include the value 0.
Again, the P-value does not directly determine the confidence interval. This statement cannot be concluded based on the given information.
c. A 99% confidence interval for m will include the value 1.
This statement is possible but cannot be concluded with certainty without more information. The P-value alone does not determine the confidence interval. However, a lower P-value suggests stronger evidence against the null hypothesis and may be indicative of a narrower confidence interval.
d. A 99% confidence interval for m will include the value 0.
This statement is also possible but cannot be concluded based on the given information. Similar to option c, the P-value alone does not determine the confidence interval.
To determine the correct answer, we would need to calculate the actual confidence interval using the data obtained from the sample. The given P-value of 0.022 suggests that there is strong evidence against the null hypothesis, but it does not directly provide information about the confidence interval.