Question

Based on information from The Denver Post, a random sample of ?1 = 12 winter days in Denver gave a sample mean pollution index of ?̅1 = 43. Previous studies show that ?1 = 21. For Englewood (a suburb of Denver), a random sample ?2 = 14 winter days gave a sample mean pollution index of ?̅2 = 36. Previous studies show that ?2 = 15. Assume the pollution index is normally distributed in both Englewood and Denver. Do these data indicate that the mean population pollution index of Englewood is different (either way) from that of Denver in the winter. Use a 1% level of significance.

a) What is the level of significance? State the null and alternative hypotheses.

b) What sampling distribution will you use? Compute the sample test statistic and correspondingz or t value as appropriate.

c) Find or estimate the P-value.

d) Based on your answers in parts (a) through (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ??

e) Interpret your conclusion in the context of the problem.

Answer 1

The provided sample means are shown below:

Also, the provided population standard deviations are:

and the sample sizes are

.

(a) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: ​

Ha: ​

The level of significance is 1%.

(b) This corresponds to a two-tailed test, for which a z-test for two population means, with known population standard deviations will be used.

Test Statistics

The z-statistic is computed as follows:

(c) The p-value is p = P(t<0.963) = 0.3355

(d) Decision about the null hypothesis

Based on the information provided, the significance level is α=0.01, and the critical value for a two-tailed test is zc​=2.58.

The rejection region for this two-tailed test is R={z:∣z∣>2.58}

Since it is observed that ∣z∣=0.963≤zc​=2.58, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: p=0.3355, and since p=0.3355≥0.01, it is concluded that the null hypothesis is not rejected.

(e)

Since there is insufficient evidence to reject the null hypothesis at 5% level of significance, we conclude that the population mean pollution index of both Denver and Englewood is the same.

Graphically

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