A humane society claims that less than 33 % of U.S. households own a dog. In a r

Question

A humane society claims that less than 33 % of U.S. households own a dog. In a random sample of 406 U.S. households, 154 say they own a dog. At = 0.01 , is there enough evidence to support the society's claim?

Write the claim mathematically and identify H0 and Ha.

Find the critical value(s) and identify the rejection region(s).

Find the standardized test statistic.

Decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

What is (are) the critical value(s)?

The critical value(s) is/are .

(Use a comma to separate answers as needed. Round to two decimal places as needed.)

In order to determine the critical values, determine the level of significance .

Next determine the type of test being conducted. Since the alternative hypothesis contains the less-than inequality symbol, the test being conducted is left-tailed.

The critical value is

Now identify the rejection region. The graph on the right shows the rejection region shaded in blue.

The critical value is negative ___, and the rejection region is z ____.

The standardized test statistic z is given by the following formula, where p is the sample proportion.

Determine the number of households out of the total sampled say that they have a dog.

Reject the null hypothesis if the standardized test statistic is in the rejection region. Fail to reject the null hypothesis if the standardized test statistic is not in the rejection region.

Use this information to draw the appropriate conclusion about the claim that less than 30 % of U.S. households own a dog.

Explanation / Answer

Here, n=sample size=406,

p=sample proportion of U.S. households own a dog=154/406=0.3793103

P=population proportion of U.S. households own a dog

We have to test here H0:P>=0.33 Vs Ha:P<0.33 .

Here, = 0.01

Critical Value=Z/2=Z0.005=2.575829 .......From standard normal distribution table

We reject H0 if Z<-Z0.005=2.575829.

Test statistic for testing above hypothesis is ,

Z0=(p-p0)/sqrt(p0(1-p0)/n)

=(0.3793103-0.33)/sqrt(0.33*0.67/406)

=2.113033

Since Z0> -Z0.005, we fail to reject H0 at 1% level of significance & conclude that greater than or equal to 33 % of U.S. households own a dog.

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