Answered

Get the information you need with the help of IDNLearn.com's expert community. Get comprehensive answers to all your questions from our network of experienced experts.

The owner of a computer company claims that the proportion of defective computer chips produced at plant A is higher than the proportion of defective chips produced by plant B.

A quality control specialist takes a random sample of 80 chips from plant A and finds 12 defective chips. Another sample of 90 chips from plant B reveals 10 defective chips.

Let [tex]\( p_A \)[/tex] be the true proportion of defective chips from plant A and [tex]\( p_B \)[/tex] be the true proportion of defective chips from plant B. The P-value for the significance test is 0.225.

Which of the following is the correct conclusion for this test of the hypotheses [tex]\( H_0: p_A - p_B = 0 \)[/tex] and [tex]\( H_a: p_A - p_B \ \textgreater \ 0 \)[/tex] at the [tex]\(\alpha = 0.05\)[/tex] level?

A. The owner should reject the null hypothesis since [tex]\( 0.225 \ \textgreater \ 0.05 \)[/tex]. There is sufficient evidence that the proportion of defective computer chips is significantly greater at plant A.
B. The owner should reject the null hypothesis since [tex]\( 0.225 \ \textgreater \ 0.05 \)[/tex]. There is insufficient evidence that the proportion of defective computer chips is significantly greater at plant A.
C. The owner should fail to reject the null hypothesis since [tex]\( 0.225 \ \textgreater \ 0.05 \)[/tex]. There is sufficient evidence that the proportion of defective computer chips is significantly greater at plant A.
D. The owner should fail to reject the null hypothesis since [tex]\( 0.225 \ \textgreater \ 0.05 \)[/tex]. There is insufficient evidence that the proportion of defective computer chips is significantly greater at plant A.

Sagot :

GinnyAnsweridn
To determine the correct conclusion for this hypothesis test, let's walk through the steps involved:

1. Hypotheses Setting:
- Null Hypothesis [tex]\( H_0 \)[/tex]: [tex]\( p_A - p_B = 0 \)[/tex], meaning there is no difference in the proportions of defective computer chips between plant [tex]\( A \)[/tex] and plant [tex]\( B \)[/tex].
- Alternative Hypothesis [tex]\( H_1 \)[/tex]: [tex]\( p_A - p_B > 0 \)[/tex], meaning the proportion of defective computer chips at plant [tex]\( A \)[/tex] is greater than at plant [tex]\( B \)[/tex].

2. Level of Significance ([tex]\(\alpha\)[/tex]):
- The significance level [tex]\(\alpha\)[/tex] is given as 0.05.

3. P-value:
- The p-value from the test is given as 0.225.

4. Decision Rule:
- To decide whether to reject the null hypothesis, we compare the p-value to the level of significance.
- If the p-value is less than [tex]\(\alpha\)[/tex] ([tex]\( p \leq \alpha \)[/tex]), we reject the null hypothesis.
- If the p-value is greater than [tex]\(\alpha\)[/tex] ([tex]\( p > \alpha \)[/tex]), we fail to reject the null hypothesis.

5. Comparison:
- Here, the p-value is 0.225.
- The level of significance [tex]\(\alpha\)[/tex] is 0.05.
- Since the p-value (0.225) is greater than [tex]\(\alpha\)[/tex] (0.05), we fail to reject the null hypothesis.

6. Conclusion:
- By failing to reject the null hypothesis, it means there is insufficient evidence to support the claim that the proportion of defective computer chips is significantly greater at plant [tex]\( A \)[/tex].

Therefore, the correct conclusion is:

The owner should fail to reject the null hypothesis since [tex]\( 0.225 > 0.05 \)[/tex]. There is insufficient evidence that the proportion of defective computer chips is significantly greater at plant [tex]\( A \)[/tex].

Hence, the correct option is:

The owner should fail to reject the null hypothesis since [tex]\( 0.225 > 0.05 \)[/tex]. There is insufficient evidence that the proportion of defective computer chips is significantly greater at plant [tex]\( A \)[/tex].
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for trusting IDNLearn.com with your questions. Visit us again for clear, concise, and accurate answers.