A study is performed in a large southern town to determine whether the average amount spent on food per four person family in the town is significantly different from the national average. A random sample of the weekly grocery bills of two-person families in this town is given in the file P08 19.xdsk. Assume the national average amount spent on food for a four person family is $$ \$ 150 $$. Family 100 Weekly food expense - $$ \$ 15,933.24 $$ a. The null hypothesis for this situation $$ \mu=150 $$ The alternative hypothesis for this situation $$ \mu \neq 150 $$ b. Is the sample evidence statistically significant? Yes If so, at what significance levels can you reject the null hypothesis? $$ 1 \%, 5 \% $$ and $$ 10 \% $$ c. For which values of the sample mean (he, average weekly grocery bill) would you reject the null hypothesis at the $$ 1 \% $$ significance level? Round your answers to one decimal place, if necessary. At the $$ 1 \% $$ significance level, if $$ X $$ is lower than $$ \qquad $$ or higher than $$ \qquad $$ , we would reject the null hypothesis. For which values of the sample mean would you reject the null hypothesis at the $$ 10 \% $$ level? Round your answers to one decimal place, if necessary. At the $$ 10 \% $$ significance level, if $$ X $$ is lower than $$ \qquad $$ or higher than $$ \qquad $$ , we would reject the null hypothesis. Display seting

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To solve this problem, we will follow these steps: 1. **Identify the known conditions**: - National average for a four-person family: $$ \mu_0 = 150 $$ - Sample size: $$ n = 100 $$ - Sample mean (from the provided data): $$ \bar{X} = 15,933.24 $$ 2. **Formulate the hypotheses**: - Null hypothesis: $$ H_0: \mu = 150 $$ - Alternative hypothesis: $$ H_a: \mu \neq 150 $$ 3. **Determine if the sample evidence is statistically significant**: - We will need to calculate the test statistic and compare it to the critical values for the significance levels of $$ 1\% $$, $$ 5\% $$, and $$ 10\% $$. 4. **Calculate the critical values for the significance levels**: - For a two-tailed test, we will find the critical values for $$ \alpha = 0.01 $$ and $$ \alpha = 0.10 $$. ### Step 1: Calculate the test statistic The test statistic for a sample mean is calculated using the formula: $$ z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}} $$ However, we need the sample standard deviation $$ \sigma $$ to compute this. Since we don't have that information, we will assume a hypothetical standard deviation for the sake of this example. Let's denote it as $$ s $$. ### Step 2: Determine critical values For a two-tailed test: - At $$ 1\% $$ significance level, the critical z-values are approximately $$ \pm 2.576 $$. - At $$ 10\% $$ significance level, the critical z-values are approximately $$ \pm 1.645 $$. ### Step 3: Calculate the rejection regions We will calculate the rejection regions for both significance levels. 1. **At the $$ 1\% $$ significance level**: - The rejection regions are: $$ \bar{X} < 150 - 2.576 \cdot \frac{s}{\sqrt{100}} \quad \text{or} \quad \bar{X} > 150 + 2.576 \cdot \frac{s}{\sqrt{100}} $$ 2. **At the $$ 10\% $$ significance level**: - The rejection regions are: $$ \bar{X} < 150 - 1.645 \cdot \frac{s}{\sqrt{100}} \quad \text{or} \quad \bar{X} > 150 + 1.645 \cdot \frac{s}{\sqrt{100}} $$ ### Step 4: Finalize the calculations Since we don't have the actual standard deviation $$ s $$, we cannot compute the exact numerical values for the rejection regions. However, if you provide the standard deviation or the sample data, I can perform the calculations to find the exact values for the rejection regions. Would you like to proceed with a hypothetical standard deviation, or do you have the actual data to calculate it? To determine if the average grocery spending is significantly different from the national average: 1. **Hypotheses**: - Null ($$ H_0 $$): The average is \$150. - Alternative ($$ H_a $$): The average is not \$150. 2. **Significance Levels**: - The sample evidence is statistically significant at $$ 1\% $$, $$ 5\% $$, and $$ 10\% $$ levels. 3. **Rejection Regions**: - **At $$ 1\% $$**: Reject $$ H_0 $$ if the sample mean is less than $$ 150 - 2.576 \cdot \frac{s}{10} $$ or greater than $$ 150 + 2.576 \cdot \frac{s}{10} $$. - **At $$ 10\% $$**: Reject $$ H_0 $$ if the sample mean is less than $$ 150 - 1.645 \cdot \frac{s}{10} $$ or greater than $$ 150 + 1.645 \cdot \frac{s}{10} $$. Note: To find exact values, the sample standard deviation $$ s $$ is needed.

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