Question

QUESTION 2. Billy’s Fizzy Flavours Inc. wants to estimate the average volume per can of Crazy 88 Cola with 90% confidence. (a) In practice, the soft drink producer does not know the population mean or population standard deviation of the volume per can. However, for the purpose of this question we will assume the volume per can is normally distributed with a mean of 353 mL and a standard deviation of 2 mL. Use this information and the Excel procedure demonstrated in Lab #3 to generate 50 independent random samples of size n = 6 cans from the production line, and compute the confidence intervals for the population mean using a 90% confidence level. In computing the confidence intervals, assume that you do not know the population standard deviation. Note: The complete spreadsheet must be included as part of your PDF file submission. Use the landscape orientation, not the default portrait orientation, to ensure all columns appear on each page. (b) Prior to generating the random samples, how many of the 50 confidence intervals in part (a) did you expect to contain the population mean of 353 mL? Explain. (c) According to your output in part (a), how many of the 50 confidence intervals successfully captured the population mean of 353 mL? Will this always match the value in part (b)? Explain. (d) Write down the six (6) observations you obtained for Sample #1 (rounded to two decimal places). Use these observations to show how, by hand, how the first confidence interval was computed. Interpret the result. (e) When we compute confidence intervals using the z-based formula, the length of every interval is the same. However, when we compute confidence intervals using the t-based formula, the length of every interval is different. Why?

Answer 1

Apologies, but it seems that you have provided a specific task related to generating random samples and conducting statistical analysis in Excel. As a text-based AI model, I'm unable to directly interact with files or generate spreadsheets.

However, I can provide you with the necessary guidance and explanations to help you complete the task. Let's go through each part of the question:

(a) To generate random samples and compute confidence intervals in Excel, you can follow these steps:

1. In Excel, create a column for each sample of size 6. You should have 50 columns, each containing 6 randomly generated values from a normal distribution with a mean of 353 mL and a standard deviation of 2 mL.

2. For each sample, calculate the sample mean.

3. Calculate the sample standard deviation for each sample.

4. Use the appropriate formula to calculate the confidence interval for each sample mean at a 90% confidence level, considering the t-distribution since the population standard deviation is unknown.

• The formula for the confidence interval is: Sample Mean ± t * (Sample Standard Deviation / √n), where t is the critical value from the t-distribution for the desired confidence level and n is the sample size.

5. Repeat steps 2-4 for all 50 samples to obtain 50 confidence intervals.

(b) Prior to generating the random samples, you would expect approximately 90% of the confidence intervals (or 45 out of 50) to contain the population mean of 353 mL. This expectation is based on the fact that a 90% confidence level implies that 90% of the confidence intervals constructed from random samples will capture the true population mean.

(c) After generating the random samples and calculating the confidence intervals, you can count how many of the 50 intervals successfully captured the population mean of 353 mL. This number may or may not be exactly 45. Due to the random nature of sampling, the actual proportion of intervals that capture the population mean may vary from the expected value. The sample results may be close to the expected value but may not match it exactly due to sampling variability.

(d) Since I don't have access to the specific random samples generated, I can't provide you with the observations for Sample #1 or show the computation of the confidence interval by hand. However, you can follow the steps mentioned in part (a) to generate the random sample for Sample #1, calculate the sample mean, sample standard deviation, and use the t-based formula to compute the confidence interval.

(e) When computing confidence intervals using the z-based formula, the length of every interval is the same because the z-distribution has a fixed standard deviation. However, when using the t-based formula, the length of each interval can differ because the t-distribution takes into account the sample size and the variability of the sample mean. As the sample size increases, the t-distribution approaches the z-distribution, and the lengths of the intervals become more consistent.

Remember, to complete this task, you will need to generate the random samples and perform the calculations in Excel. Make sure to consult relevant statistical resources or consult your instructor for any specific instructions or guidelines provided for this assignment.