Answer :
To determine if there is sufficient evidence at the 5% significance level to conclude that the mean weight of US males is more than 180 pounds, we will perform a hypothesis test. Here’s how you can perform the test:
Define the Hypotheses:
- Null Hypothesis (H0): [tex]\mu = 180[/tex] pounds (The mean weight of US males is 180 pounds.)
- Alternative Hypothesis (H1): [tex]\mu > 180[/tex] pounds (The mean weight of US males is more than 180 pounds.)
Collect the Sample Data:
- Sample weights: {190, 185, 188, 195, 170, 175, 190, 195, 190, 170}
Calculate the Sample Mean and Standard Deviation:
Sample mean ([tex]\bar{x}[/tex]) is calculated as:
[tex]\bar{x} = \frac{190 + 185 + 188 + 195 + 170 + 175 + 190 + 195 + 190 + 170}{10} = 184.8 \text{ pounds}[/tex]Sample standard deviation (s) is calculated with:
[tex]s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}[/tex]After calculating, [tex]s \approx 10.92 \text{ pounds}[/tex]
Conduct the t-Test:
- Compute the t-statistic using:
[tex]t = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{184.8 - 180}{10.92/\sqrt{10}} \approx 1.389[/tex]
- Compute the t-statistic using:
Find the P-value:
- Using a t-distribution table or calculator with degrees of freedom:[tex]df = n - 1 = 9[/tex]
- A t-statistic of 1.389 corresponds to a p-value of approximately 0.108.
Compare P-value and Significance Level:
- At [tex]\alpha = 0.05[/tex], since the p-value (0.108) is greater than [tex]\alpha[/tex], we fail to reject the null hypothesis.
Conclusion:
- There is not enough evidence to conclude that the mean weight of US males is more than 180 pounds at the 5% significance level.
The closest specified p-value in the given options to our calculated p-value (0.108) is option d) 0.1284.
