Question

Let x be a random variable representing dividend yield of bank stocks. We may assume that x has a normal distribution with σ = 1.9%. A random sample of 10 bank stocks gave the following yields (in percents). 5.7 4.8 6.0 4.9 4.0 3.4 6.5 7.1 5.3 6.1 The sample mean is x bar = 5.38%. Suppose that for the entire stock market, the mean dividend yield is μ = 4.7%. Do these data indicate that the dividend yield of all bank stocks is higher than 4.7%? Use α = 0.01. (a) What is the level of significance? (Enter a number.) State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? H0: μ = 4.7%; H1: μ ≠ 4.7%; two-tailed H0: μ > 4.7%; H1: μ = 4.7%; right-tailed H0: μ = 4.7%; H1: μ > 4.7%; right-tailed H0: μ = 4.7%; H1: μ < 4.7%; left-tailed (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The standard normal, since we assume that x has a normal distribution with known σ. The Student's t, since n is large with unknown σ. The standard normal, since we assume that x has a normal distribution with unknown σ. The Student's t, since we assume that x has a normal distribution with known σ. Compute the z value of the sample test statistic. (Enter a number. Round your answer to two decimal places.) (c) Find (or estimate) the P-value. (Enter a number. Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (Select the correct graph.) The graph of a bell-shaped distribution has a horizontal axis labeled at -3, -2, -1, 0, 1, 2 and 3. The curve enters the viewing window just above the axis at -3, reaches a peak at 0, and exits the viewing window just above the axis at 3. The area under the curve from about -1.1 to the left is shaded. The graph of a bell-shaped distribution has a horizontal axis labeled at -3, -2, -1, 0, 1, 2 and 3. The curve enters the viewing window just above the axis at -3, reaches a peak at 0, and exits the viewing window just above the axis at 3. The areas under the curve from about -1.1 to the left and from about 1.1 to the right are shaded. The graph of a bell-shaped distribution has a horizontal axis labeled at -3, -2, -1, 0, 1, 2 and 3. The curve enters the viewing window just above the axis at -3, reaches a peak at 0, and exits the viewing window just above the axis at 3. The area under the curve from about 1.1 to the left is shaded. The graph of a bell-shaped distribution has a horizontal axis labeled at -3, -2, -1, 0, 1, 2 and 3. The curve enters the viewing window just above the axis at -3, reaches a peak at 0, and exits the viewing window just above the axis at 3. The area under the curve from about 1.1 to the right is shaded. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) State your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the average yield for bank stocks is higher than that of the entire stock market. There is insufficient evidence at the 0.01 level to conclude that the average yield for bank stocks is higher than that of the entire stock market.

Answer 1

Solution:

Given:

x be a random variable representing dividend yield of bank stocks and x has a normal distribution with σ = 1.9%.

A random sample of 10 bank stocks gave the following yields (in percents). 5.7 4.8 6.0 4.9 4.0 3.4 6.5 7.1 5.3 6.1

Sample Mean =

μ = 4.7%

We have to test if the dividend yield of all bank stocks is higher than 4.7% or not.

Part a)  What is the level of significance?

The level of significance = α = 0.01.

State the null and alternate hypotheses:

Since we have to test if the dividend yield of all bank stocks is higher than 4.7%, this is right tailed test.

Thus correct option is:

H₀: μ = 4.7%; H₁: μ > 4.7%; right-tailed

Part b) What sampling distribution will you use?

The standard normal, since we assume that x has a normal distribution with known σ.

Compute the z value of the sample test statistic

Part c) Find (or estimate) the P-value.

P-value = P( Z > z test statistic value)

P-value = P( Z > 1.13 )

P-value =1 - P( Z < 1.13 )

Look in z table for z = 1.1 and 0.03 and find area.

P( Z < 1.13) = 0.8708

Thus

P-value =1 - P( Z < 1.13 )

P-value =1 - 0.8708

P-value = 0.1292

Sketch the sampling distribution and show the area corresponding to the P-value.

Part d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

Since P-value = 0.1292 > 0.01 level of significance , we fail to reject null hypothesis H₀.

Thus data is not statistically significant at 0.01 level of significance.

Thus correct option is:

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

Part e) State your conclusion in the context of the application.

There is insufficient evidence at the 0.01 level to conclude that the average yield for bank stocks is higher than that of the entire stock market.