A random sample of 51 of the largest companies in the united states gave the following P/e ratios
11 35 19 13 15 21 40 18 60 72 9 20
29 53 16 26 21 14 21 27 10 12 47 14
33 14 18 17 20 19 13 25 23 27 5 16
8 49 44 20 27 8 19 12 31 67 51 26
19 18 32
Use a calculator with mean and sample standard deviation keys to verify that xbar = 25.2 and s=15.5
find a 99 % confidence interval for the P/E population mean of all large companies
Bank One merged with JP morgan had a P/E of 12, AT7t had a P/E of 72 Disney had a P/E of 24 Examine the confidence intervals in b and c how would you describe the stocks at the time the sample was taken
To calculate the mean (x̄) and sample standard deviation (s) using a calculator with mean and sample standard deviation keys, follow these steps:
Step 1: Enter the data into the calculator.
Enter the given P/E ratios into the calculator: 11, 35, 19, 13, 15, 21, 40, 18, 60, 72, 9, 20, 29, 53, 16, 26, 21, 14, 21, 27, 10, 12, 47, 14, 33, 14, 18, 17, 20, 19, 13, 25, 23, 27, 5, 16, 8, 49, 44, 20, 27, 8, 19, 12, 31, 67, 51, 26, 19, 18, 32.
Step 2: Calculate the mean (x̄).
Use the mean (x̄) key on your calculator to obtain the mean of the data set: x̄ = 25.2.
Step 3: Calculate the sample standard deviation (s).
Use the sample standard deviation (s) key on your calculator to obtain the sample standard deviation of the data set: s = 15.5.
Now, to find a 99% confidence interval for the P/E population mean of all large companies, use the following formula:
Confidence Interval = x̄ ± (z * (s / √n))
Where:
x̄ = sample mean
z = z-score corresponding to the desired confidence level (for 99%, z ≈ 2.58)
s = sample standard deviation
n = sample size
Plugging in the values:
x̄ = 25.2
s = 15.5
n = 51
z ≈ 2.58 (for 99% confidence level)
Confidence Interval = 25.2 ± (2.58 * (15.5 / √51))
Calculating the Confidence Interval:
Confidence Interval = 25.2 ± (2.58 * 2.16)
Confidence Interval = 25.2 ± 5.57
Therefore, the 99% confidence interval for the P/E population mean of all large companies is approximately (19.63, 30.77).
Considering the stocks at the time the sample was taken:
- Bank One's P/E of 12 falls within the confidence interval, suggesting it is not statistically different from the population mean.
- AT&T's P/E of 72 is outside the confidence interval, indicating that it is above the population mean.
- Disney's P/E of 24 falls within the confidence interval, suggesting it is not statistically different from the population mean.