Question

Nationally, about 11% of the total U.S. wheat crop is destroyed each year by hail.† An insurance company is studying wheat hail damage claims in a county in Colorado. A random sample of 16 claims in the county reported the percentage of their wheat lost to hail.

17 7 9 13 13 20 13 11 5 8 26 21 13 8 12 4

The sample mean is x = 12.5%. Let x be a random variable that represents the percentage of wheat crop in that county lost to hail. Assume that x has a normal distribution and σ = 5.0%. Do these data indicate that the percentage of wheat crop lost to hail in that county is different (either way) from the national mean of 11%? Use α = 0.01.

(a) What is the level of significance?

State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

H0: μ ≠ 11%; H1: μ = 11%; two-tailed

H0: μ = 11%; H1: μ > 11%; right-tailed

H0: μ = 11%; H1: μ ≠ 11%; two-tailed

H0: μ = 11%; H1: μ < 11%; 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 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.

The Student's t, since n is large with unknown σ.

Compute the z 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.)

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

(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 hail damage to wheat crops in the county in Colorado differs from the national average.

There is insufficient evidence at the 0.01 level to conclude that the average hail damage to wheat crops in the county in Colorado differs from the national average.

Answer 1

The statistical software output for this problem is:

Hence,

a) Level of significance = 0.01

H0: μ = 11%; H1: μ ≠ 11%; two-tailed

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

Sample test statistic = 1.20

c) P-value = 0.2301

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

e) There is insufficient evidence at the 0.01 level to conclude that the average hail damage to wheat crops in the county in Colorado differs from the national average.