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
To make the confidence interval, you need to find the mean and the standard deviation for the data.
Since this is a small sample and you don't know the population standard deviation, you will use "t" instead of "z".
Using a t-table you will have to find the value for .01 2tailed test.
Use that value in your formula to calculate the confidence interval.
b) there are several assumptions that you will make including randomness and normality of the data.
Question 2 This deals with % /proportions.
You will use a 1 proportion z to calculate this interval. 95% gives you 5% divided by 2 tails =2.5% or .025. Use the formula to find the standard deviation. You will need p-hat, q-hat and n.
For the last question, there is a formula to find sample size. You are given a .02 margin of error. Assume the confidence interval is still 95%.
Question 1: Production Rates
(a) To calculate the confidence interval for the average number of items produced per hour, we can use the formula:
CI = x̄ ± t * (s/√n)
Where:
- x̄ is the sample mean
- t is the critical value from the t-distribution for the desired confidence level and degrees of freedom (in this case, 99% confidence level and 7 degrees of freedom)
- s is the sample standard deviation
- n is the sample size
First, let's calculate the sample mean (x̄):
142 + 175 + 162 + 158 + 190 + 154 + 160 + 185 = 1236
x̄ = 1236 / 8 = 154.5
Next, we need to calculate the sample standard deviation (s). To do this, we can use the following formula:
s = √((Σ(xi - x̄)^2) / (n-1))
Where:
- Σ(xi - x̄)^2 is the sum of the squared differences between each value and the sample mean
- n is the sample size
For the given data:
(xi - x̄)^2: (142-154.5)^2 + (175-154.5)^2 + (162-154.5)^2 + (158-154.5)^2 + (190-154.5)^2 + (154-154.5)^2 + (160-154.5)^2 + (185-154.5)^2 = 2490
s = √(2490 / (8-1)) = √(2490 / 7) ≈ 8.579
Next, we need to find the t-value for the desired confidence level (99%) and degrees of freedom (n-1 = 7). The t-value can be found using a t-table or a statistical calculator. For a 99% confidence level and 7 degrees of freedom, the t-value is approximately 3.499.
Now we can calculate the confidence interval:
CI = 154.5 ± 3.499 * (8.579 / √8) ≈ 154.5 ± 11.294
The 99% confidence interval for the average number of items produced per hour is approximately (143.206, 165.794). This means that we are 99% confident that the true average number of items produced per hour is within this range.
(b) The assumption made in order to answer part (a) is that the production rates are normally distributed, and the sample is random and representative of the population.
Question 2: Customer Satisfaction
(a) To construct a confidence interval for the proportion of unsatisfied customers, we can use the formula:
CI = p̂ ± z * √((p̂(1-p̂))/n)
Where:
- p̂ is the sample proportion
- z is the critical value from the standard normal distribution for the desired confidence level (in this case, 95% confidence level)
- n is the sample size
First, let's calculate the sample proportion (p̂):
p̂ = 45 / 200 = 0.225
Next, we need to find the z-value for the desired confidence level (95%). The z-value can be found using a standard normal distribution table or a statistical calculator. For a 95% confidence level, the z-value is approximately 1.96.
Now we can calculate the confidence interval:
CI = 0.225 ± 1.96 * √((0.225(1-0.225))/200) ≈ 0.225 ± 0.060
The 95% confidence interval for the proportion of unsatisfied customers is approximately (0.165, 0.285). This means that we are 95% confident that the true proportion of unsatisfied customers is within this range.
(b) To calculate the sample size necessary to estimate the proportion to within 2% of the population proportion, we can use the formula:
n = (z^2 * p̂(1-p̂)) / (E^2)
Where:
- z is the critical value from the standard normal distribution for the desired confidence level (in this case, 95% confidence level)
- p̂ is the estimated proportion (we can use 0.5 as a conservative estimate when no prior estimate is known)
- E is the desired margin of error in decimal form (in this case, 2% = 0.02)
Plugging in the values:
n = (1.96^2 * 0.5(1-0.5)) / (0.02)^2 ≈ 2401
Therefore, a sample size of approximately 2401 would be necessary to estimate the proportion to within 2% of the population proportion.