Question

Asked Jul 15, 2019

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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.

16 | 6 | 8 | 9 | 13 | 22 | 13 | 13 |

7 | 12 | 23 | 19 | 11 | 9 | 13 | 4 |

The sample mean is x = 12.4%. 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?

*H*_{0}: μ ≠ 11%; *H*_{1}: μ = 11%; two-tailed*H*_{0}: μ = 11%; *H*_{1}: μ ≠ 11%; two-tailed *H*_{0}: μ = 11%; *H*_{1}: μ < 11%; left-tailed*H*_{0}: μ = 11%; *H*_{1}: μ > 11%; right-tailed

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

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.)

(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?

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

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

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

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

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.

Compute the

(c) Find (or estimate) the

(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.

Step 1

** (a) Level of significance:**

It is given that to test whether the percentage of wheat crop lost to hail in that country is different from the national mean of 11%, the level of significance *α* = 0.01.

That is, here the level of significance (*α*) is **0.01.**

**Null and alternative hypotheses:**

*Null hypothesis:*

*H*_{0}: *µ* = 11%

That is, the percentage of wheat crop lost to hail in that country is equal to the national mean of 11%.

*Alternative hypothesis:*

*H*_{1}: *µ* ≠ 11%

That is, the percentage of wheat crop lost to hail in that country is different from the national mean of 11%,

Since, the alternative hypothesis states that *µ* ≠ 11%, it is a **two-tailed test.**

Step 2

**(b) Sampling distribution of the test:**

It is given that *σ* = 5.0% and *X* has a normal distribution.

Since we assume that *X* has a normal distribution with known σ, the sampling distribution for the test is **standard** **normal distribution ( z-test)**.

**Test statistic for z-test:**

Here, the sample mean, *x*-bar is 12.4%.

Population mean, *µ* is 11%.

Population standard deviation, *σ* = 5.0%.

Sample size, *n* is 16.

The test statistic for *z*-test is calculated as **1.12** from the calculations given below.

Step 3

**(c) Computation of P-value:**

The *P*-value for the *z*-test can be obtained using the excel formula,** “**=2*Z.TEST(A1:A16,11,...

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