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 verify that the mean (x̄) is 25.2 and the sample standard deviation (s) is 15.5, you can use a calculator with mean and sample standard deviation keys.
1. Enter the given data into your 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
2. Calculate the mean (x̄) using the mean key:
x̄ = (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) / 51 = 25.2
3. Calculate the sample standard deviation (s) using the sample standard deviation key:
s = √([(11-25.2)² + (35-25.2)² + ... + (32-25.2)²] / (51-1)) = 15.5
Now that you have verified the mean and standard deviation, you can proceed to find a 99% confidence interval for the P/E population mean of all large companies.
To find the confidence interval, you need to use the following formula:
Confidence Interval = x̄ ± Z * (s / √n)
Where:
x̄ is the sample mean
Z is the Z-score associated with the desired confidence level (in this case, 99%)
s is the sample standard deviation
n is the sample size
1. Determine the Z-score associated with a 99% confidence level. You can look up this value in a Z-score table or use a calculator. For a 99% confidence level, the Z-score is approximately 2.576.
2. Calculate the confidence interval:
Confidence Interval = 25.2 ± 2.576 * (15.5 / √51)
Calculating the square root of 51 (√51), we find that it is approximately 7.14.
Confidence Interval = 25.2 ± 2.576 * (15.5 / 7.14) = 25.2 ± 5.62
Therefore, the 99% confidence interval for the P/E population mean of all large companies is (19.58, 30.82), rounded to two decimal places.
Regarding the stocks at the time the sample was taken:
- Bank One merged with JP Morgan had a P/E of 12, which is below the lower bound of the confidence interval. This means the P/E ratio of Bank One merged with JP Morgan was relatively low compared to the population mean at that time.
- AT&T had a P/E of 72, which is above the upper bound of the confidence interval. This indicates that AT&T had a relatively high P/E ratio compared to the population mean at that time.
- Disney had a P/E of 24, which is within the confidence interval. This suggests that Disney had a P/E ratio relatively close to the population mean at that time.