Scores on a certain IQ test are known to have a mean of . A random sample of 100 students attend a series of coaching classes before taking the test. Let u be the population mean IQ score that would occur if every student took the coaching classes. The classes are successful if u > 100. A test is made of the hypotheses Hv0 : u =100 versus Hv1 : u>100. Consider three possible conclusions: (i) The classes are successful. (ii) The classes are not successful. (iii) The classes might not be successful.

Question

Which of the 3 conclusions is best if Hv0 is rejected?

Answer 1

When discussing hypothesis testing, it's important to clarify what each conclusion means in relation to the hypotheses:

• Ho (null hypothesis): \( u = 100 \) (the population mean IQ score for students who took the coaching classes is 100)
• H1 (alternative hypothesis): \( u > 100 \) (the population mean IQ score for students who took the coaching classes is greater than 100)

If you reject the null hypothesis \( H0 \), you are concluding that there is sufficient evidence to support the alternative hypothesis \( H1 \), which suggests that the coaching classes have had a positive effect, and thus the classes are successful.

Now let's analyze the three possible conclusions:

1. The classes are successful: This conclusion aligns with rejecting the null hypothesis. If you have enough evidence to assert that \( u > 100 \), it implies that the coaching classes have effectively improved student IQ scores, so this would be a strong and correct conclusion upon rejection.

2. The classes are not successful: This conclusion is contrary to what would be suggested by rejecting the null hypothesis. If you have rejected \( H0 \), saying the classes are not successful contradicts the statistical evidence and would generally be wrong.

3. The classes might not be successful: This conclusion reflects uncertainty and does not align with a decisive rejection of the null hypothesis. If you reject \( H0 \), you provide evidence that suggests that the mean IQ is indeed greater than 100, which means the coaching classes are likely successful.

Given the context of rejecting the null hypothesis \( H0 \), the best conclusion is:

(i) The classes are successful.

This conclusion is supported by rejecting \( H0 \) and indicates that there is statistical evidence to suggest the coaching classes positively impact the IQ scores.

Answer 2

Scores on a certain IQ test are known to have a mean of 100. A random sample of 74 students attend a series of coaching classes before taking the test. Let u be the population mean IQ score that would occur if every student took the coaching classes. The classes are successful if u > 100. A test is made of the hypotheses Hv0 : u =100 versus Hv1 : u>100. Consider three possible conclusions: (i) The classes are successful. (ii) The classes are not successful. (iii) The classes might not be successful.
Which of the 3 conclusions is best if Hv0 is rejected?

Answer 3

In the context provided, we have the following hypotheses for testing the effectiveness of the coaching classes on IQ scores:

• Null Hypothesis (\(H_0\)): \(u = 100\) (the population mean IQ score of students who took the coaching classes is 100)
• Alternative Hypothesis (\(H_1\)): \(u > 100\) (the population mean IQ score of students who took the coaching classes is greater than 100)

If the null hypothesis \(H_0\) is rejected, it indicates that there is statistically significant evidence suggesting that the true mean IQ score for the students who attended the coaching classes is greater than 100.

Now let's analyze the three possible conclusions you provided:

1. The classes are successful: This conclusion aligns with rejecting the null hypothesis. If you reject \(H_0\) in favor of \(H_1\), you are asserting that there is sufficient evidence to say that the coaching classes have improved the IQ scores above the baseline mean of 100.

2. The classes are not successful: This conclusion contradicts the rejection of \(H_0\). If you reject \(H_0\), this would imply that the coaching classes are indeed successful, making this conclusion incorrect.

3. The classes might not be successful: Similar to the previous point, this conclusion shows uncertainty and does not align with the action of rejecting \(H_0\). If you reject the null hypothesis, it means you are reasonably confident that the mean IQ is greater than 100, so saying that they might not be successful goes against the presented evidence.

Given that you rejected \(H_0\), the best conclusion among the options provided is:

(i) The classes are successful.

This conclusion is the most appropriate because rejecting \(H_0\) supports the indication that the coaching classes have a positive effect on the IQ scores.

Answer 4

Assume that the classes are not successful. Is it possible to make a Type II error? Explain.

[Yes or no?], a type 2 error [is or isnt] possible. The classes are not successful when the null hypothesis is [true or false?]

Answer 5

To understand your question regarding Type II errors, we first need to clarify what a Type II error is:

• Type II Error (\( \beta \)): This occurs when we fail to reject the null hypothesis (\( H_0 \)) when it is actually false. In the context of your test, it would mean concluding that the classes are not successful (by failing to reject \( H_0 \)) when in fact they are successful (meaning \( u > 100 \)).

Now, let's address your question:

• If we assume that the classes are not successful, it means that the true population mean IQ score (\( u \)) is equal to or less than 100.

• In this case, if the null hypothesis \( H_0: u = 100 \) is true, then it is not possible to make a Type II error because a Type II error pertains to failing to reject a false null hypothesis. Here, we assume \( H_0 \) is true, which means there's no error to make concerning the acceptance of \( H_0 \).

Now to answer your prompts directly:

• [Yes or no?]: No, a Type II error isn't possible.
• The classes are not successful when the null hypothesis is true.

In summary: Under the assumption that the classes are not successful, you wouldn't make a Type II error, because a Type II error occurs when you fail to reject a false null hypothesis, whereas in this case, you have assumed the null hypothesis is actually true.