A local grocery store wants to estimate the mean daily number of gallons of
milk sold to customers. Assume the number of gallons sold follows the normal
distribution, with a population standard deviation of 5.10 gallons. A random
sample of 60 days shows that the mean daily number of gallons sold is 10.00.
1) What is the point estimate of the population mean?
2) Compute a 99% confidence interval for the population mean.
1. the point estimate of the population mean is 10 gallons.
2. [(10-2.575*5.10/sqrt((60)), 10+ 2.575*5.10/sqrt((60))])
= [8.3, 11.7]
To answer these questions, we can use the concept of point estimates and confidence intervals.
1) Point Estimate of the Population Mean:
The point estimate of the population mean is the sample mean. In this case, the sample mean daily number of gallons sold is given as 10.00. Therefore, the point estimate of the population mean is 10.00 gallons.
2) 99% Confidence Interval for the Population Mean:
A confidence interval is a range of values in which the true population parameter (in this case, the mean daily number of gallons sold) is likely to fall. To compute a 99% confidence interval for the population mean, we will need the sample mean, sample size, and the population standard deviation.
Given:
Sample mean (x̄) = 10.00 gallons
Sample size (n) = 60 days
Population standard deviation (σ) = 5.10 gallons
To compute the confidence interval, we can use the following formula:
Confidence Interval = x̄ ± Z * (σ/√n)
Where:
- x̄ represents the sample mean
- Z represents the z-value, which corresponds to the desired level of confidence (99% in this case)
- σ represents the population standard deviation
- n represents the sample size
To find the Z value corresponding to a 99% confidence level, we can use a standard normal distribution table or a statistical calculator. The Z value for a 99% confidence level is approximately 2.576.
Plugging the values into the formula, we get:
Confidence Interval = 10.00 ± 2.576 * (5.10/√60)
Now, we can solve for the confidence interval:
Confidence Interval = 10.00 ± 2.576 * (5.10/√60) = (7.852, 12.148)
The 99% confidence interval for the mean daily number of gallons sold is (7.852, 12.148) gallons. This means we can be 99% confident that the true population mean falls within this range.
1) The point estimate of the population mean is the sample mean, which is 10.00 gallons.
2) To compute a 99% confidence interval for the population mean, we can use the formula:
Confidence interval = sample mean ± (critical value) × (standard deviation / √sample size)
Since the sample size is 60, the standard deviation is 5.10 gallons, and we want a 99% confidence level, we need to find the critical value associated with a 99% confidence level.
The critical value can be found using a Z-table or a statistical software. For a 99% confidence level, the critical value is approximately 2.576.
Now, we can calculate the confidence interval:
Confidence interval = 10.00 ± (2.576) × (5.10 / √60)
Confidence interval = 10.00 ± (2.576) × (0.658)
Confidence interval = 10.00 ± 1.695
Therefore, the 99% confidence interval for the population mean is approximately (8.305, 11.695) gallons.