Question

1a. Suppose x has a distribution with μ = 21 and σ = 16.

(a) If a random sample of size n = 39 is drawn, find μ_x, σ_x and P(21 ≤ x ≤ 23). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(21 ≤ x bar ≤ 23) =

(b) If a random sample of size n = 55 is drawn, find μ_x, σ_x and P(21 ≤ x ≤ 23). (Round σ_x to two decimal places and the probability to four decimal places.)

μx =

σx =

P(21 ≤ x bar ≤ 23) 1b.Find P(69 ≤ x ≤ 74). (Round your answer to four decimal places.) 1c. Bill Alther is a zoologist who studies Anna's hummingbird (Calypte anna).† Suppose that in a remote part of the Grand Canyon, a random sample of six of these birds was caught, weighed, and released. The weights (in grams) were as follows. 3.7 2.9 3.8 4.2 4.8 3.1 The sample mean is x = 3.75 grams. Let x be a random variable representing weights of hummingbirds in this part of the Grand Canyon. We assume that x has a normal distribution and σ = 0.92 gram. Suppose it is known that for the population of all Anna's hummingbirds, the mean weight is μ = 4.55 grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.55 grams? Use α = 0.05. (a) What is the level of significance? (b) What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find (or estimate) the P-value. (Round your answer to four decimal places.) 1D. The price to earnings ratio (P/E) is an important tool in financial work. A random sample of 14 large U.S. banks (J. P. Morgan, Bank of America, and others) gave the following P/E ratios†. 24 16 22 14 12 13 17 22 15 19 23 13 11 18 The sample mean is x≈ 17.1. Generally speaking, a low P/E ratio indicates a "value" or bargain stock. Suppose a recent copy of a magazine indicated that the P/E ratio of a certain stock index is μ = 19. Let x be a random variable representing the P/E ratio of all large U.S. bank stocks. We assume that x has a normal distribution and σ = 3.6. Do these data indicate that the P/E ratio of all U.S. bank stocks is less than 19? Use α = 0.01. (a) What is the level of significance? (b)What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find (or estimate) the P-value. (Round your answer to four decimal places.)

Answer 1

1)a) = 21

       = = 16/ = 2.56

P(21 < < 23)

= P((21 - )/() < ( - )/() < (23 - )/())

= P((21 - 21)/2.56 < Z < (23 - 21)/2.56)

= P(0 < Z < 0.78)

= P(Z < 0.78) - P(Z < 0)

= 0.7823 - 0.5000

= 0.2823

b) = 21

       = = 16/ = 2.16

P(21 < < 23)

= P((21 - )/() < ( - )/() < (23 - )/())

= P((21 - 21)/2.16 < Z < (23 - 21)/2.16)

= P(0 < Z < 0.93)

= P(Z < 0.93) - P(Z < 0)

= 0.8238 - 0.5000

= 0.3238

1c)a) Level of significance = 0.05

The test statistic z = ()/()

                             = (3.75 - 4.55)/(0.92/)

                              = -2.13

c) P-value = P(Z < -2.13)

                 = 0.0166

1D)a) Level of significance = 0.01

b) The test statistic z = ()/()

                                  = (17.1 - 19)/(3.6/)

                                  = -1.97

c) P-value = P(Z < -1.97)

                 = 0.0244