Question

The monthly expenditures on food by single adults living in one neighborhood of Los Angeles are normally distributed with a mean of $410 and a standard deviation of $75. Determine the percentage of samples of size 9 that have mean expenditures within $20 of the population mean expenditure of $410.

Answer 1

To calculate the percentage of samples of size 9 that have mean expenditures within $20 of the population mean, we can use the t-distribution and calculate the probability.

Step-by-step explanation:

To determine the percentage of samples of size 9 that have mean expenditures within $20 of the population mean of $410, we can calculate the probability using the normal distribution.

We know that the mean is $410 and the standard deviation is $75. Since the sample size is less than 30, we can use the t-distribution. We need to find the probability that the sample mean falls within $20 of the population mean.

Using the t-distribution, we can calculate the critical t-value for a 95% confidence level with 8 degrees of freedom (sample size minus 1). From the t-distribution table, the critical t-value is approximately 2.306.

Next, we calculate the standard error of the mean by dividing the standard deviation by the square root of the sample size: $75/sqrt(9) = $25. Finally, we can use the t-distribution formula to calculate the probability:

t = (sample mean - population mean) / standard error of the mean = (410 - 410) / 25 = 0

Since the calculated t-value is 0, which is less than the critical t-value of 2.306, we can conclude that there is a 95% probability that the sample mean falls within $20 of the population mean.

Answer 2

0.5763

Step-by-step explanation:

We need to estimate the standard error of the mean.

Standard error of the mean = standard deviation of the original distribution/√sample size

Standard error of the mean = 75/√9 = 25

Now we can use this Standard error to estimate z as follows:

Z = (x – mean)/standard deviation

We want the mean expenditures within $20, so x – mean = 20 and -20

Z = (20)/25

Z = 0.8

Z = (-20)/25

Z = -0.8

Using a Z table we can find probability

P (-0.8<z<0.8)= 0.5763