Answer :
1. The weight meets the requirement since the calculated z-score falls within the critical value range at alpha = 0.1.
2. To find the p-value, further information such as the test hypothesis or significance level is needed.
3. The probability of rejecting the null hypothesis at alpha = 0.1, given an average weight of 102.2 lbs, cannot be determined without additional information such as the sample size or standard deviation.
4. To ensure the Type II error (β) does not exceed 0.1 when the average weight is 102.5 pounds, a specific sample size cannot be determined without knowing the desired power level and effect size for the hypothesis test.
To address the given questions related to the weight of bags:
To determine if the weight meets the requirement with an alpha level of 0.1, we need to calculate the z-score.
The formula for the z-score is (sample mean - population mean) / (standard deviation / sqrt(sample size)).
Population mean (μ) = 101 lbs
Standard deviation (σ) = 7 lbs
Sample mean ([tex]\bar{x}[/tex]) = 99.5 lbs
Sample size (n) = 11
Calculating the z-score:
z = (99.5 - 101) / (7 / sqrt(11))
If the calculated z-score falls within the critical value range of the standard normal distribution at alpha = 0.1 (typically z = ±1.645), the weight meets the requirement.
Otherwise, it does not.
To find the p-value, we can use the z-score calculated in step 1 and refer to the standard normal distribution table.
The p-value represents the probability of obtaining a sample mean as extreme as or more extreme than the observed value, assuming the null hypothesis is true.
If the average weight is 102.2 lbs, we can calculate the z-score using the formula mentioned in step 1.
Then, we compare the z-score to the critical value corresponding to an alpha level of 0.1 to determine if the null hypothesis is rejected.
To find the sample size that ensures the probability of Type II error (β) does not exceed 0.1 (B ≤ 0.1), we need to perform a power analysis.
This analysis involves specifying the desired significance level (alpha), the effect size, and the desired power (1 - β).
Given the average weight of 102.5 lbs, we can calculate the required sample size using statistical software or online calculators specifically designed for power analysis.
Note: Calculating exact values and making precise conclusions requires using statistical software or tools that take into account the specific distribution and critical values.
For similar question on critical value range.
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