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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].
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