Question 1 Production Rates

A manufacturing company wishes to estimate the number of items that its workforce can produce on average each hour now that they have a new machine. The factory examined the records for a random sample of 8 hours over the past month. The hourly production rates for these 8 hours were:
142 175 162 158 190 154 160 185
(a) Calculate by hand and interpret the 99 percent confidence interval for the average number of items produced per hour.
(b) What assumption did you make in order to answer part (a)?
Question 2 Customer Satisfaction
(a) An insurance company wants to estimate the proportion of people unsatisfied with their new telephone help service. A survey of 200 callers revealed 45 were unsatisfied with the service. Construct by hand a 95% confidence interval for the proportion of unsatisfied customers.
(b) If the company wanted to estimate the sample proportion to within 2% of the population proportion, what sample size would be necessary assuming no prior estimate of the proportion is known?

Question 1 Production Rates:

(a) To calculate the 99% confidence interval for the average number of items produced per hour, you can follow these steps:

1. Calculate the sample mean (x̄) by finding the average of the hourly production rates:
x̄ = (142 + 175 + 162 + 158 + 190 + 154 + 160 + 185) / 8

2. Calculate the sample standard deviation (s) by using the formula:
s = √((Σ(xi - x̄)²) / (n - 1)),
where Σ denotes the sum of, xi represents each individual hourly production rate, x̄ is the sample mean, and n is the sample size.

3. Determine the critical value (z*) for a 99% confidence level. You can use a standard normal distribution table to find the z-value or utilize statistical software.

4. Calculate the margin of error (E) using the formula:
E = z* * (s / √n),
where z* represents the critical value, s is the sample standard deviation, and n is the sample size.

5. Construct the confidence interval using the formula:
Confidence Interval = x̄ ± E.

Once you have the values for x̄, s, z*, and E, you can calculate the confidence interval.

(b) The assumption made in part (a) is that the sample data follows a normal distribution. This assumption is necessary to use a z-distribution for constructing the confidence interval. Additionally, it assumes that the sample is random and independent of each other.

Question 2 Customer Satisfaction:

(a) To construct a 95% confidence interval for the proportion of unsatisfied customers, you can follow these steps:

1. Calculate the sample proportion (p̂) by dividing the number of unsatisfied customers by the total sample size:
p̂ = 45 / 200.

2. Determine the critical value (z*) for a 95% confidence level. You can refer to a standard normal distribution table or utilize statistical software.

3. Calculate the margin of error (E) using the formula:
E = z* * √((p̂(1 - p̂)) / n),
where z* is the critical value, p̂ represents the sample proportion, and n is the sample size.

4. Construct the confidence interval using the formula:
Confidence Interval = p̂ ± E.

Once you have the values for p̂, z*, and E, you can calculate the confidence interval.

(b) To determine the sample size necessary to estimate the sample proportion within 2% of the population proportion, you can use the following formula:

n = (z*² * (p̂ * (1 - p̂))) / (E²),
where n represents the required sample size, z* is the critical value, p̂ represents the proportion estimate (assumed to be 0.5 if unknown), and E is the maximum tolerable margin of error (0.02 in this case).

Plug in the values for z*, p̂, and E into the formula to calculate the necessary sample size.