To test the claim that the proportion of people who own cats is smaller than 40% at the 0.01 significance level, follow these steps:
### Step 1: State the Hypotheses
We need to determine the null hypothesis [tex]\( H_0 \)[/tex] and the alternative hypothesis [tex]\( H_1 \)[/tex]:
- Null hypothesis: [tex]\( H_0: p = 0.4 \)[/tex]
- Alternative hypothesis: [tex]\( H_1: p < 0.4 \)[/tex]
This is a left-tailed test because we are testing if the sample proportion is less than the population proportion.
### Step 2: Given Information and Sample Data
- Sample proportion ([tex]\( \hat{p} \)[/tex]): [tex]\( 0.31 \)[/tex]
- Sample size ([tex]\( n \)[/tex]): [tex]\( 100 \)[/tex]
- Population proportion ([tex]\( p \)[/tex]): [tex]\( 0.4 \)[/tex]
- Significance level ([tex]\( \alpha \)[/tex]): [tex]\( 0.01 \)[/tex]
### Step 3: Calculate the Standard Error
The standard error (SE) of the proportion is calculated using the formula:
[tex]\[ SE = \sqrt{\frac{p(1-p)}{n}} \][/tex]
From the calculations, we have:
[tex]\[ SE = 0.049 \][/tex]
### Step 4: Calculate the Test Statistic (z)
The test statistic [tex]\( z \)[/tex] is calculated using the formula:
[tex]\[ z = \frac{\hat{p} - p}{SE} \][/tex]
From the calculations, we have:
[tex]\[ z = -1.837 \][/tex]
### Step 5: Calculate the p-value
The p-value corresponding to the test statistic [tex]\( z \)[/tex] for a left-tailed test is the area under the standard normal curve to the left of [tex]\( z \)[/tex].
From the calculations, we have:
[tex]\[ p\text{-value} = 0.0331 \][/tex]
### Step 6: Compare the p-value with the Significance Level
We compare the p-value to the significance level [tex]\( \alpha \)[/tex]:
- If the p-value < [tex]\( \alpha \)[/tex], we reject the null hypothesis.
- If the p-value ≥ [tex]\( \alpha \)[/tex], we fail to reject the null hypothesis.
From the calculations, we have:
[tex]\[ p\text{-value} = 0.0331 \][/tex]
[tex]\[ \alpha = 0.01 \][/tex]
Since the p-value (0.0331) is greater than the significance level (0.01), we fail to reject the null hypothesis.
### Conclusion
Based on the sample data and the given significance level:
- Hypotheses:
[tex]\[
\begin{array}{l}
H_0: p = 0.4 \\
H_1: p < 0.4
\end{array}
\][/tex]
- The test is: left-tailed
- Sample proportion: [tex]\( 0.31 \)[/tex]
- Test statistic: [tex]\( -1.837 \)[/tex]
- p-value: [tex]\( 0.0331 \)[/tex]
- Decision: Fail to reject the null hypothesis
### Final Interpretation
There is not enough evidence at the 0.01 significance level to reject the null hypothesis. Therefore, we do not have sufficient evidence to support the claim that the proportion of people who own cats is smaller than 40%.
