Question

Part 1) Independent random samples of professional football and basketball players gave the following information. Assume that the weight distributions are mound-shaped and symmetric.

Weights (in lb) of pro football players: x₁; n₁ = 21

248	261	255	251	244	276	240	265	257	252	282
256	250	264	270	275	245	275	253	265	270

Weights (in lb) of pro basketball players: x₂; n₂ = 19

203	200	220	210	191	215	221	216	228	207
225	208	195	191	207	196	183	193	201

(a) Use a calculator with mean and standard deviation keys to calculate x₁, s₁, x₂, and s₂. (Round your answers to one decimal place.)

x1 =
s1 =
x2 =
s2 =

(b) Let ?₁ be the population mean for x₁ and let ?₂ be the population mean for x₂. Find a 99% confidence interval for ?₁ ? ?₂. (Round your answers to one decimal place.)

lower limit:

upper limit: Part 2) The U.S. Geological Survey compiled historical data about Old Faithful Geyser (Yellowstone National Park) from 1870 to 1987. Let x1 be a random variable that represents the time interval (in minutes) between Old Faithful eruptions for the years 1948 to 1952. Based on 9720 observations, the sample mean interval was x1 = 65.0 minutes. Let x2 be a random variable that represents the time interval in minutes between Old Faithful eruptions for the years 1983 to 1987. Based on 25,691 observations, the sample mean time interval was x2 = 71.2 minutes. Historical data suggest that ?1 = 8.77 minutes and ?2 = 12.34 minutes. Let ?1 be the population mean of x1 and let ?2 be the population mean of x2. (a) Compute a 95% confidence interval for ?1 – ?2. (Use 2 decimal places.) lower limit: upper limit: Part 3) On the Navajo Reservation, a random sample of 214 permanent dwellings in the Fort Defiance region showed that 66 were traditional Navajo hogans. In the Indian Wells region, a random sample of 147 permanent dwellings showed that 26 were traditional hogans. Let p1 be the population proportion of all traditional hogans in the Fort Defiance region, and let p2 be the population proportion of all traditional hogans in the Indian Wells region. (a) Find a 99% confidence interval for p1 – p2. (Use 3 decimal places.) lower limit: upper limit:

Answer 1

1) a) x1 = 259.7

s1 = 11.9

x2 = 205.8

s2 = 12.7

b) DF = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^2/n2)^2/(n2 - 1))

          = ((11.9)^2/21 + (12.7)^2/19)^2/(((11.9)^2/21)^2/20 + ((12.7)^2/19)^2/18)

          = 37

At 99% confidence interval the critical value is t_{0.005, 37} = 2.715

The 99% confidence interval is

(x1 - x2) +/- t_{0.005, 37} * sqrt(s1^2/n1 + s2^2/n2)

= (259.7 - 205.8) +/- 2.715 * sqrt((11.9)^2/21 + (12.7)^2/19)

= 53.9 +/- 10.6

= 43.3, 64.5

lower limit = 43.3

Upper limit = 64.5

2)a) At 95% confidence interval the critical value is z_0.025 = 1.96

(x1 - x2) +/- z_0.025 *

= (65 - 71.2) +/- 1.96 * sqrt((8.77)^2/9720 + (12.34)^2/25691)

= -6.2 +/- 0.23

= -6.43, -5.97

lower limit = -6.43

upper limit = -5.97

3) = 66/214 = 0.308

   = 26/147 = 0.177

The pooled sample proportion(P) = ( * n1 + * n2)/(n1 + n2)

                                                      = (0.308 * 214 + 0.177 * 147)/(214 + 147)

                                                      = 0.255

SE = sqrt(P * (1 - P) * (1/n1 + 1/n2))

      = sqrt(0.255 * (1 - 0.255) * (1/214 + 1/147))

      = 0.047

At 99% confidence interval the critical value is z_0.005 = 2.58

The 99% confidence interval is

+/- z_0.005 * SE

= (0.308 - 0.177) +/- 2.58 * 0.047

= 0.131 +/- 0.121

= 0.010, 0.252