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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

Ask by Griffin Sandoval. in the United States
Mar 06,2025

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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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Beyond the Answer

The context of hypothesis testing in statistics is fundamental to discerning trends and averages in various domains. In your scenario, you are investigating if the average weekly grocery bill of families in a southern town significantly deviates from the national average of $150 for four-person families. Crafting null and alternative hypotheses sets the stage for statistical analysis, allowing you to utilize techniques like t-tests or z-tests to determine significance versus the threshold of variance in data. To determine the cutoff values for the sample mean \( X \) at different significance levels, you would generally refer to the critical values from the z-distribution or t-distribution table. For the 1% significance level (two-tailed) in a typical scenario with a large sample size, the critical values might range approximately from \( X < 148.0 \) or \( X > 152.0 \). Meanwhile, at the 10% significance level, the cutoffs could be roughly \( X < 149.0 \) or \( X > 151.0 \). These intervals inform you when the evidence is strong enough to reject the null hypothesis, paving the way for more tailored conclusions about spending habits in your specific community!

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