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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 production at plant A and determines that there are 12 defective chips. The specialist then takes a random sample of 90 chips from production at plant B and determines that there are 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 [tex]P[/tex]-value for this 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.



Answer :

GinnyAnswer
Let's solve the problem step by step to arrive at the correct conclusion for the significance test.

### Step 1: Understand the Hypotheses
We are testing the claim that the proportion of defective chips is higher at plant A than at plant B. This can be stated in terms of the following hypotheses:

- Null hypothesis (\(H_0\)): \( p_A - p_B = 0 \) (There is no difference in the proportion of defective chips between plant A and plant B)
- Alternative hypothesis (\(H_1\)): \( p_A - p_B > 0 \) (Plant A has a higher proportion of defective chips than plant B)

### Step 2: Given Data
- Sample size from plant A (\(n_A\)): 80
- Number of defective chips from plant A: 12
- Sample size from plant B (\(n_B\)): 90
- Number of defective chips from plant B: 10

### Step 3: Significance Level
- Significance level (\(\alpha\)): 0.05

### Step 4: P-Value
It is given that the p-value for this test is 0.225.

### Step 5: Compare P-Value with Significance Level
To determine whether we reject or fail to reject the null hypothesis, we need to compare the p-value to the significance level (\(\alpha\)):

- If the p-value \(\leq \alpha\), we reject the null hypothesis.
- If the p-value \(> \alpha\), we fail to reject the null hypothesis.

Given that the p-value (0.225) is greater than the significance level (0.05):

[tex]\[0.225 > 0.05 \][/tex]

### Step 6: Draw Conclusion
Since the p-value is greater than the significance level, we fail to reject the null hypothesis. This means that there is not enough evidence to support the claim that the proportion of defective computer chips is significantly greater at plant A than at plant B.

Thus, the correct conclusion is:

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

So 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].

This corresponds to option 4.