Question

A researcher claims that the proportion of cars with manual transmission is less than 10%. To test this claim, a survey checked 1000 randomly selected cars. Of those cars, 95 had a manual transmission. The following is the setup for the hypothesis test: H0:p=0.10 Ha:p<0.10 In this example, the p-value was determined to be 0.299. Come to a conclusion and interpret the results for this hypothesis test for a proportion (use a significance level of 5%) Select the correct answer below: The decision is to reject the Null Hypothesis. The conclusion is that there is enough evidence to support the claim. The decision is to fail to reject the Null Hypothesis. The conclusion is that there is not enough evidence to support the claim.

Answer 1

The calculated value Z = 22.22 > 1.96 at 5% level of significance

Therefore the null hypothesis is rejected.

The conclusion is that there is not enough evidence to support the claim

Explanation:

Step 1:-

A researcher claims that the proportion of cars with manual transmission is less than 10%.

P = 10% = 0.10

Given sample size n= 1000

given the Sample proportion is p = 0.299

Null hypothesis: H₀: P =0.10

Alternative hypothesis :Ha:p<0.10

Level of significance α=0.05

Step 2:-

The test of hypothesis
`Z = \frac{p-P}{\sqrt{(PQ)/(n ) } }`

put all values , we get

`Z = \frac{0.299-0.10}{\sqrt{(0.10X0.90)/(1000 ) } }`

on calculation , we get

`Z = (0.199)/(0.0094) = 22.22`

The 5 % level of significance of tabulated value = 1.96

Conclusion:-

The calculated value Z = 22.22 > 1.96 at 5% level of significance

Therefore the null hypothesis is rejected.

The conclusion is that there is not enough evidence to support the claim

Answer 2

The decision is to fail to reject the Null Hypothesis. The conclusion is that there is not enough evidence to support the claim.

Explanation:

A one-proportion z-test can be used to determine whether the proportion of cars with manual transmission is less than 10%.

The hypothesis is defined as:

H₀: The proportion of cars with manual transmission is 10%, i.e. p = 0.10.

Hₐ: The proportion of cars with manual transmission is less than 10%, i.e. p < 0.10.

The information provided is:

`X=95\\n=1000\\\alpha =0.05\\p-value=0.299`

The test statistic is:

`z=\frac{\hat p-p}{\sqrt{(p(1-p))/(n)}}`

Decision rule:

If the p-value of the test is less than the significance level (α) then the null hypothesis will be rejected or else accepted.

The p-value = 0.299 > α = 0.05.

The null hypothesis was failed to rejected at 5% level of significance.

Conclusion:

As the null hypothesis was not rejected it can be concluded that the proportion of cars with manual transmission is 10%.

Thus, the correct option is:

"The decision is to fail to reject the Null Hypothesis. The conclusion is that there is not enough evidence to support the claim."