Question

A carpet company advertises that it will deliver your carpet within 15 days of purchase. A sample of 49 past customers is taken. The average delivery time in the sample was 16.2 days. The standard deviation of the population (0) is known to be 5.6 days. a. State the null and alternative hypotheses for a test to determine if their advertisement is legitimate. b. Using the critical value approach, test the hypotheses. Let a = .05. C. Using the p-value approach, test the hypotheses at the 5% level of significance.

Answer 1

Solution:

The provided sample mean is Xˉ=16.2 and the known population standard deviation is \σ=5.6, and the sample size is n=49.

(1) Null and Alternative Hypotheses

Claim: A carpet company will deliver your carpet within 15 days of purchase i.e μ≤15

The following null and alternative hypotheses need to be tested:

Ho: μ≤15 ...(claim)

Ha: μ>15

This corresponds to a right-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is

zc​=1.64.

The rejection region for this right-tailed test is R={z:z>1.64}

(3) Test Statistics

The z-statistic is computed as follows:

X - до 16.2 - 15 5.6/49 = 1.5 с/уп

(4) Decision about the null hypothesis

Since it is observed that z=1.5≤zc​=1.64, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.0668, and since p=0.0668≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is sufficient evidence to claim that the population mean μ is less than or equal to 15, at the 0.05 significance level.

Confidence Interval

The 95% confidence interval is 14.632<μ<17.768.

Graphically

Z-Test Results: t-stats = 1.5, p-value =0.0668 0.40 0.35 0.30 0.25 - 0.20 - 0.15 0.10 - 0.05 0.00 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Test Hypothesis:

Ho: μ≤15 ...(claim)

Ha: μ>15

Critical Value approach:

The rejection region for this right-tailed test is R={z:z>1.64}

Since it is observed that z=1.5≤zc​=1.64, it is then concluded that the null hypothesis(claim) is not rejected.

P value approach:

Using the P-value approach: The p-value is p=0.0668, and since p=0.0668≥0.05, it is concluded that the null hypothesis(claim) is not rejected.

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