Which of the following statements are true?
(A) In testing the hypothesis for the equality of two population proportions, the acceptance region for the null hypothesis will be wider than the confidence interval for two population proportions, assuming the same value of alpha.
(B) A 98% confidence interval estimate for the difference between two population proportions is always symmetrical with zero at the center of the interval.
(C) The test of hypothesis for the equality of two population means based on independent samples with 9 observations from each population will have 16 degrees of freedom.
(D) A quality control manager claims that the proportion of defective units produced by machine I (Population 1) is less than the proportion produced by machine II (Population 2). The correct null hypothesis is Ho: p1 – p2 >= 0.
Question 11 options:
A and B only
C and D only
B and D only
A and C only
To determine which statements are true, let's evaluate each statement separately:
(A) In testing the hypothesis for the equality of two population proportions, the acceptance region for the null hypothesis will be wider than the confidence interval for two population proportions, assuming the same value of alpha.
To understand this, we need to know that the acceptance region for the null hypothesis in hypothesis testing is what enables us to fail to reject the null hypothesis. In this case, when testing the equality of two population proportions, the null hypothesis assumes that the proportions are equal. The acceptance region is usually determined based on the significance level (alpha).
On the other hand, a confidence interval for two population proportions estimates a range of values within which the true difference between the proportions is expected to fall. It provides a range of plausible values.
To answer this statement, we need to compare the width of the acceptance region in hypothesis testing with the width of the confidence interval. In general, the acceptance region is narrower than the confidence interval. This is because hypothesis testing relies on more specific criteria for accepting or rejecting the null hypothesis, while the confidence interval aims to provide a range of plausible values, allowing for more variability.
Therefore, statement (A) is NOT true.
(B) A 98% confidence interval estimate for the difference between two population proportions is always symmetrical, with zero at the center of the interval.
To determine the correctness of this statement, we need to understand what a confidence interval is for the difference between two population proportions. A confidence interval estimates a range of plausible values within which the true difference between the proportions is expected to fall. In general, the confidence interval is centered around the point estimate (the observed difference) with a margin of error on either side.
However, the claim in this statement is not entirely correct. A confidence interval is not always perfectly symmetrical, especially when the sample sizes or proportions are very different. The shape of the confidence interval is affected by various factors, such as sample sizes, proportions, and the assumption of independence between the populations.
Therefore, statement (B) is NOT true.
(C) The test of hypothesis for the equality of two population means based on independent samples with 9 observations from each population will have 16 degrees of freedom.
To evaluate this statement, we need to understand how degrees of freedom (df) are calculated for hypothesis testing with independent samples. In this case, the degrees of freedom for the t-test are calculated as the sum of the individual degrees of freedom for each sample.
For independent samples, the degrees of freedom calculation involves subtracting the total number of groups from the total number of observations. In this statement, we have two independent samples with 9 observations each. So the total number of observations is 9 + 9 = 18.
To calculate the degrees of freedom, we subtract the total number of groups (2) from the total number of observations (18): df = 18 - 2 = 16.
Therefore, statement (C) is TRUE.
(D) A quality control manager claims that the proportion of defective units produced by machine I (Population 1) is less than the proportion produced by machine II (Population 2). The correct null hypothesis is Ho: p1 – p2 >= 0.
To determine the correctness of this statement, we need to understand how to formulate the null and alternative hypotheses in hypothesis testing.
In this case, the manager claims that the proportion of defective units by machine I is less than the proportion produced by machine II. When formulating the null hypothesis (Ho), we assume equality or no difference between the proportions. The alternative hypothesis (Ha) suggests a specific direction of difference, in this case, it would be Ha: p1 - p2 < 0, indicating that the proportion by machine I is less than machine II.
The null hypothesis (Ho) should be formed as the opposite of the alternative hypothesis, which in this case would be Ho: p1 - p2 >= 0.
Therefore, statement (D) is TRUE.
In summary, the correct answer is:
(B) and (D) only.