Question

The price of a share of stock divided by the company's estimated future earnings per share is called the P/E ratio. High P/E ratios usually indicate "growth" stocks, or maybe stocks that are simply overpriced. Low P/E ratios indicate "value" stocks or bargain stocks. 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

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean x and sample standard deviation s. (Round your answers to one decimal place.)

x =
s =

(b) Find a 90% confidence interval for the P/E population mean μ of all large U.S. companies. (Round your answers to one decimal place.)

lower limit
upper limit

(c) Find a 99% confidence interval for the P/E population mean μ of all large U.S. companies. (Round your answers to one decimal place.)

lower limit
upper limit

Answer 1

	Values ( X )	Σ ( Xi- X̅ )2
	11	200.9732
	35	96.5012
	19	38.1492
	13	148.2672
	15	103.5612
	21	17.4432
	40	219.7362
	18	51.5022
	60	1212.6762
	72	2192.4402
	9	261.6792
	20	26.7962
	29	14.6192
	53	774.1472
	16	84.2082
	26	0.6782
	21	17.4432
	14	124.9142
	21	17.4432
	27	3.3252
	10	230.3262
	12	173.6202
	47	476.2652
	14	124.9142
	33	61.2072
	14	124.9142
	18	51.5022
	17	66.8552
	20	26.7962
	19	38.1492
	13	148.2672
	25	0.0312
	23	4.7372
	27	3.3252
	5	407.0912
	16	84.2082
	8	295.0322
	49	567.5592
	44	354.3242
	20	26.7962
	27	3.3252
	8	295.0322
	19	38.1492
	12	173.6202
	31	33.9132
	67	1749.2052
	51	666.8532
	26	0.6782
	19	38.1492
	18	51.5022
	32	46.5602
Total	1284	11969.4142

Part a)

Mean X̅ = Σ Xi / n
X̅ = 1284 / 51 = 25.2

Sample Standard deviation S_X = √ ( (Xi - X̅ )² / n - 1 )
S_X = √ ( 11969.44 / 51 -1 ) = 15.5

Part b)

Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.1 /2, 51- 1 ) = 1.676
25.1765 ± t(0.1/2, 51 -1) * 15.4722/√(51)
Lower Limit = 25.1765 - t(0.1/2, 51 -1) 15.4722/√(51)
Lower Limit = 21.5454 ≈ 21.4
Upper Limit = 25.1765 + t(0.1/2, 51 -1) 15.4722/√(51)
Upper Limit = 28.8076 ≈ 28.8
90% Confidence interval is ( 21.5 , 28.8 )

Part c)
Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.01 /2, 51- 1 ) = 2.678
25.1765 ± t(0.01/2, 51 -1) * 15.4722/√(51)
Lower Limit = 25.1765 - t(0.01/2, 51 -1) 15.4722/√(51)
Lower Limit = 19.3745 ≈ 19.4
Upper Limit = 25.1765 + t(0.01/2, 51 -1) 15.4722/√(51)
Upper Limit = 30.9785 ≈ 31.0
99% Confidence interval is ( 19.4 , 31.0 )