Question
A major DVD rental chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5000 of the 20000 households in the area are equipped with DVD players. It conducts a telephone poll of 300 randomly selected households in the area and finds that 96 have DVD players. The rental chain's conclusion from the hypothesis test using a 5% level of significance is:
to open a new store.
not to open a new store.
to delay opening a new store until additional evidence is collected.
We cannot tell what the decision should be from the information given.
to open a new store.
not to open a new store.
to delay opening a new store until additional evidence is collected.
We cannot tell what the decision should be from the information given.
Answers
GPT-4o mini
To determine the decision of the DVD rental chain regarding opening a new store, we can conduct a hypothesis test.
1. **Define the hypotheses**:
- Null hypothesis (\(H_0\)): \(p \leq 0.25\) (where \(p\) is the proportion of households with DVD players, implying that 5000 out of 20000 households have DVD players)
- Alternative hypothesis (\(H_a\)): \(p > 0.25\)
2. **Collect the data**:
- Sample size (\(n\)) = 300
- Number of households with DVD players in the sample = 96
- Sample proportion (\(\hat{p}\)) = \(96/300 = 0.32\)
3. **Calculate the test statistic**:
- The test statistic for a proportion can be calculated using the formula:
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]
where \(p_0\) is the hypothesized proportion (0.25).
Substituting in the values:
- \(p_0 = 0.25\)
- \(\hat{p} = 0.32\)
- \(n = 300\)
The standard error (SE) can be calculated as:
\[
SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.25 \times 0.75}{300}} = \sqrt{\frac{0.1875}{300}} \approx 0.025
\]
Now calculate \(z\):
\[
z = \frac{0.32 - 0.25}{0.025} = \frac{0.07}{0.025} \approx 2.8
\]
4. **Determine the critical z-value**:
- For a one-tailed test at the 5% level of significance, the critical z-value is approximately 1.645.
5. **Make a decision**:
- Since our calculated z-value of 2.8 is greater than the critical z-value of 1.645, we reject the null hypothesis.
6. **Conclusion**:
- The rental chain has sufficient evidence to conclude that more than 5000 households have DVD players. Therefore, the decision is:
**to open a new store.**
1. **Define the hypotheses**:
- Null hypothesis (\(H_0\)): \(p \leq 0.25\) (where \(p\) is the proportion of households with DVD players, implying that 5000 out of 20000 households have DVD players)
- Alternative hypothesis (\(H_a\)): \(p > 0.25\)
2. **Collect the data**:
- Sample size (\(n\)) = 300
- Number of households with DVD players in the sample = 96
- Sample proportion (\(\hat{p}\)) = \(96/300 = 0.32\)
3. **Calculate the test statistic**:
- The test statistic for a proportion can be calculated using the formula:
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]
where \(p_0\) is the hypothesized proportion (0.25).
Substituting in the values:
- \(p_0 = 0.25\)
- \(\hat{p} = 0.32\)
- \(n = 300\)
The standard error (SE) can be calculated as:
\[
SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.25 \times 0.75}{300}} = \sqrt{\frac{0.1875}{300}} \approx 0.025
\]
Now calculate \(z\):
\[
z = \frac{0.32 - 0.25}{0.025} = \frac{0.07}{0.025} \approx 2.8
\]
4. **Determine the critical z-value**:
- For a one-tailed test at the 5% level of significance, the critical z-value is approximately 1.645.
5. **Make a decision**:
- Since our calculated z-value of 2.8 is greater than the critical z-value of 1.645, we reject the null hypothesis.
6. **Conclusion**:
- The rental chain has sufficient evidence to conclude that more than 5000 households have DVD players. Therefore, the decision is:
**to open a new store.**