Question

test the claim that the proportion of people who own dogs is less than 33%. a random sample of 1000 people found that 27.9% owned dogs. do the sample data provide convincing evidence to support the claim? test the relevant hypotheses using a 5% level of significance. give answer to at least 4 decimal places.
a. what are the correct hypotheses? (select the correct symbols and use decimal values not percentages.)

based on the hypotheses, find the following:
b. test statistic =

c. p - value =

d. based on the above we choose to

e. the correct summary would be: that the proportion of people who own dogs is less than 33%.
question help:

Explanation:

Step1: Set up hypotheses

The null hypothesis \(H_0\) is a statement of no - change or equality. The alternative hypothesis \(H_1\) is the claim we are testing.
The population proportion is denoted by \(p\). The claim is that the proportion of people who own dogs is less than \(33\%=0.33\).
\(H_0:p = 0.33\)
\(H_1:p<0.33\)

Step2: Calculate the test statistic

The formula for the test statistic \(z\) in a one - sample proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
where \(\hat{p}=0.279\) (sample proportion), \(p = 0.33\) (hypothesized proportion), and \(n = 1000\) (sample size)
First, calculate the denominator \(\sqrt{\frac{p(1 - p)}{n}}=\sqrt{\frac{0.33\times(1 - 0.33)}{1000}}=\sqrt{\frac{0.33\times0.67}{1000}}=\sqrt{\frac{0.2211}{1000}}=\sqrt{0.0002211}\approx0.0149\)
Then, \(z=\frac{0.279 - 0.33}{0.0149}=\frac{- 0.051}{0.0149}\approx - 3.4228\)

Step3: Calculate the p - value

Since \(H_1:p<0.33\), the p - value is \(P(Z<z)\) where \(z\approx - 3.4228\)
Using a standard normal table or a calculator with a normal distribution function (e.g., in Excel: NORM.S.DIST(-3.4228,TRUE)), the p - value is approximately \(0.0003\)

Step4: Make a decision

The significance level \(\alpha = 0.05\)
Since the p - value (\(0.0003\)) is less than \(\alpha(0.05)\), we reject \(H_0\)

Step5: Summarize the results

We reject the null hypothesis. So, there is convincing evidence to support the claim that the proportion of people who own dogs is less than \(33\%\)

Answer:

a. \(H_0:p = 0.33\); \(H_1:p<0.33\)
b. \(-3.4228\)
c. \(0.0003\)
d. Reject \(H_0\)
e. There is convincing evidence