12) Which of the following statements is true?

a) The sampling distribution of the difference between two proportions
will always be normal.
b) When comparing two population proportions, either sample proportion
can be used as the unbiased estimate of the true population proportion.
c) If sample sizes are large enough, and if the population is large
compared to the sample, we can compare two proportions using a normal
approximation.
d) When you’re doing inference for two proportions, then number of
degrees of freedom is one less than the smaller of the two sample sizes.
e) None of these are true

D

To determine which of the statements is true, let's go through each statement one by one:

a) The statement that the sampling distribution of the difference between two proportions will always be normal is not true. The conditions for the normality of the sampling distribution depend on the sample sizes, the population proportions, and the independence of the samples.

b) This statement is not true. When comparing two population proportions, it is necessary to use the sample proportions from each sample as unbiased estimates of the true population proportions. The sample proportions are used to calculate the standard errors and construct confidence intervals or conduct hypothesis tests.

c) This statement is partially true. If the sample sizes are large enough (typically greater than 30), and if the population is large compared to the sample (usually considered as at least 10 times larger), then a normal approximation can be used to compare two proportions. However, the size of the population alone is not sufficient to determine whether a normal approximation can be used - the sample sizes play a crucial role.

d) This statement is not true. The number of degrees of freedom in inference for two proportions depends on the specific test being conducted. For most commonly used tests (e.g., the z-test or chi-square test), the degrees of freedom are not equal to one less than the smaller of the two sample sizes.

e) Since none of the statements are entirely true, the correct answer is e) None of these are true.

To determine which of the statements is true, let's analyze each statement one by one:

a) The sampling distribution of the difference between two proportions will always be normal.
To understand if this statement is true, we can consider the Central Limit Theorem (CLT). According to the CLT, when the sample size is large enough, the sampling distribution of the difference between two proportions will be approximately normal. Therefore, statement a) is true.

b) When comparing two population proportions, either sample proportion can be used as the unbiased estimate of the true population proportion.
To determine whether this statement is true, we need to understand the concept of unbiased estimates. Unbiased estimates are estimators whose expected value is equal to the true population parameter. In the case of proportions, the sample proportions can be used as unbiased estimates. Therefore, statement b) is true.

c) If sample sizes are large enough, and if the population is large compared to the sample, we can compare two proportions using a normal approximation.
To assess the validity of this statement, we need to take into account the requirements for using a normal approximation. The conditions mentioned in statement c) are correct; if the sample sizes are large enough and the population is sufficiently larger than the sample, then a normal approximation can be used to compare two proportions. Hence, statement c) is true.

d) When you’re doing inference for two proportions, the number of degrees of freedom is one less than the smaller of the two sample sizes.
The statement in d) is incorrect. Inference for comparing two proportions uses a test statistic that follows a different distribution depending on the method chosen (e.g., Z-test or chi-square test). The degrees of freedom are determined based on the specific method, not simply by subtracting one from the smaller sample size. Therefore, statement d) is false.

e) None of these are true.
Based on the analysis above, we have determined that statements a), b), and c) are true. Therefore, statement e) is false.

In conclusion, the correct statement among the options provided is:
- a) The sampling distribution of the difference between two proportions will always be normal.
- b) When comparing two population proportions, either sample proportion can be used as the unbiased estimate of the true population proportion.
- c) If sample sizes are large enough, and if the population is large compared to the sample, we can compare two proportions using a normal approximation.