Question

Let x be a random variable representing dividend yield of bank stocks. We may assume that x has a normal distribution with σ = 2.3%. 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 = 5.38%. Suppose that for the entire stock market, the mean dividend yield is μ = 4.9%. Do these data indicate that the dividend yield of all bank stocks is higher than 4.9%? 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 > 4.9%; H₁: μ = 4.9%; right-tailed H₀: μ = 4.9%; H₁: μ ≠ 4.9%; two-tailed     H₀: μ = 4.9%; H₁: μ > 4.9%; right-tailed H₀: μ = 4.9%; H₁: μ < 4.9%; left-tailed

(b)

What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

The Student's t, since we assume that x has a normal distribution with known σ. The standard normal, since we assume that x has a normal distribution with unknown σ.     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.

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.)

(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

The provided sample mean is Xˉ=5.38

and the known population standard deviation is σ=2.3,

and the sample size is n = 10.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ=4.9

Ha: μ>4.9

This corresponds to a right-tailed test, for which a z-test for one mean, with known population standard deviation, will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=.01, and the critical value for a right-tailed test is

z_c = 2.33

The rejection region for this right-tailed test is

R={z:z>2.33}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that

z = 0.66 ≤ zc ​= 2.33,

it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is p = 0.2546,

and since p = 0.2546 ≥ 0.01,

it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected.

Therefore, there is not enough evidence to claim that the population mean μ is greater than 4.9, at the .01 significance level.

hence , 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.

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