Question

A humane society claims that less than 65​% of households in a certain country own a pet. In a random sample of 500 households in that​ country, 305 say they own a pet. At α=0.01​, is there enough evidence to support the​ society's claim? Complete parts​ (a) through​ (c) below.

Answer 1

At α=0.01​, there is enough evidence to support the​ society's claim that less than 65% of households in a certain country own a pet.

Step-by-step explanation:

Null hypothesis (H0): The proportion of households in the country that own a pet is 65% or more.

Alternative hypothesis (Ha): The proportion of households in the country that own a pet is less than 65%.

Step-by-step solution:

1. Calculate the test statistic, which is the z-score.
2. Find the p-value corresponding to the test statistic.
3. Compare the p-value to the significance level (α = 0.01).
4. If the p-value is less than α, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

First, calculate the proportion of households in the sample that own a pet: 305/500 = 0.61.

Next, calculate the standard error of the proportion: sqrt((0.65(1-0.65))/500) = 0.0194.

Calculate the test statistic (z-score): (0.61-0.65)/0.0194 = -2.06.

Find the p-value corresponding to a z-score of -2.06 (using a standard normal distribution table or a calculator): p-value = 0.019.

Since the p-value (0.019) is less than the significance level (0.01), we reject the null hypothesis.

Therefore, there is enough evidence to support the humane society's claim that less than 65% of households in the country own a pet.