Consider the trash bag problem. Suppose that an independent laboratory has tested trash bags and has found that no 30-gallon bags that are currently on the market have a mean breaking strength of 50 pounds or more. On the basis of these results, the producer of the new, improved trash bag feels sure that its 30-gallon bag will be the strongest such bag on the market if the new trash bag’s mean breaking strength can be shown to be at least 50 pounds. The mean of the sample of 40 trash bag breaking strengths in Table 1.9 is x = 50.575. If we let m denote the mean of the breaking strengths of all possible trash bags of the new type and assume that s equals 1.65:
a. Calculate 95 percent and 99 percent confidence intervals for m.
b. Using the 95 percent confidence interval, can we be 95 percent confident that m is at least 50 pounds? Explain.
c. Using the 99 percent confidence interval, can we be 99 percent confident that m is at least 50 pounds? Explain.
d. Based on your answers to parts b and c, how convinced are you that the new 30-gallon trash bag is the strongest such bag on the market?
To answer these questions, we need to calculate the confidence intervals for the mean breaking strength of the new trash bag. Let's break down the steps to solve this problem:
Step 1: Calculate the standard deviation (s) of the sample
Given that the sample mean (x) is 50.575 and the sample size is 40, we can use the formula for the sample standard deviation:
s = sqrt((∑(x - x̄)^2) / (n - 1))
where x̄ is the sample mean, and n is the sample size.
Step 2: Calculate the standard error (SE) of the sample mean
The standard error measures the average amount of variation in the sample mean. It can be calculated using the formula:
SE = s / sqrt(n)
Step 3: Calculate the confidence intervals
a. For the 95 percent confidence interval:
- Determine the critical value, which corresponds to the desired confidence level. Since we want a 95 percent confidence interval, the critical value is 1.96 (for a normal distribution).
- Calculate the margin of error (ME) using the formula: ME = critical value * SE
- The confidence interval is given by: (x̄ - ME, x̄ + ME)
b. For the 99 percent confidence interval:
- Determine the critical value, which corresponds to the desired confidence level. For a 99 percent confidence interval, the critical value is 2.626.
- Calculate the margin of error (ME) using the formula: ME = critical value * SE
- The confidence interval is given by: (x̄ - ME, x̄ + ME)
Now, let's go through each part of the question:
a. Calculate 95 percent and 99 percent confidence intervals for m.
Using the values given in the problem statement, you can calculate the confidence intervals by substituting these values into the formulas mentioned above.
b. Using the 95 percent confidence interval, can we be 95 percent confident that m is at least 50 pounds? Explain.
To answer this question, we need to check if the lower bound of the 95 percent confidence interval is above 50 pounds. If it is, we can be 95 percent confident that m is at least 50 pounds.
c. Using the 99 percent confidence interval, can we be 99 percent confident that m is at least 50 pounds? Explain.
Similar to part b, we need to check if the lower bound of the 99 percent confidence interval is above 50 pounds. If it is, we can be 99 percent confident that m is at least 50 pounds.
d. Based on your answers to parts b and c, how convinced are you that the new 30-gallon trash bag is the strongest such bag on the market?
This question requires you to compare the results from parts b and c. If the 99 percent confidence interval includes values above 50 pounds, but the 95 percent confidence interval does not, you might have more confidence in the claim. However, it is subjective and depends on the specific context and level of confidence required.
By following these steps and performing the calculations, you will be able to answer the questions about the confidence intervals and evaluate the strength claim for the new trash bag.