Final answer:
The question asks how to construct a 95% confidence interval for the average weight of metal rings and how to perform hypothesis testing to determine if their weight is less than 50 ounces. A confidence interval is calculated and a one-sample t-test is used for hypothesis testing.
Step-by-step explanation:
Constructing a 95% Confidence Interval
To construct a 95% confidence interval for the average weight of the metal rings, we use the formula:
Confidence Interval = sample mean ± (critical value) × (standard deviation/√n)
For a 95% confidence level with 49 degrees of freedom, the critical value (t-score) can be found in a t-distribution table or using statistical software. Assuming it is approximately 2 (for large sample sizes, the t-distribution approaches the z-distribution), we calculate:
Confidence Interval = 49.84 ± 2 × (6.80/√50)
This gives us the range for the average weight of the metal rings made by the company.
Hypothesis Testing for Average Weight
To test the hypothesis that the average weight is less than 50 ounces, we perform a one-sample t-test:
H0: μ = 50 (null hypothesis)
H1: μ < 50 (alternative hypothesis)
We calculate the t-statistic using the formula:
t = (sample mean - population mean) / (standard deviation/√n)
t = (49.84 - 50) / (6.80/√50)
The calculated t-value will be compared against the critical t-value from a t-distribution table for 49 degrees of freedom and a 5% significance level.
If the calculated t-value is less than the critical t-value, we reject the null hypothesis and conclude that the average weight is significantly less than 50 ounces.