# Nationally, about 11% of the t

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

13 | 8 | 10 | 10 | 10 | 22 | 16 | 9 |

9 | 8 | 24 | 20 | 11 | 8 | 13 | 4 |

The sample mean is *x* = 12.2%. Let *x* be arandom variable that represents the percentage of wheat crop inthat county lost to hail. Assume that *x* has a normaldistribution and *σ* = 5.0%. Do these data indicate that thepercentage 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 aleft-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%;right-tailed *H*_{0}:*μ* = 11%; *H*_{1}: *μ*< 11%; left-tailed*H*_{0}: *μ* = 11%;*H*_{1}: *μ* ≠ 11%; two-tailed

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

The standard normal, since we assume that *x* has anormal distribution with known *σ*.The Student’s *t*,since *n* is large with unknown*σ*. The standard normal, sincewe 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. (Roundyour answer to two decimal places.)(c) Find (or estimate) the *P*-value. (Round your answer tofour decimal places.)Sketch the sampling distribution and show the area corresponding tothe *P*-value.

(d) Based on your answers in parts (a) to (c), will you reject orfail to reject the null hypothesis? Are the data statisticallysignificant at level *α*?

At the *α* = 0.01 level, we reject the null hypothesisand conclude the data are statistically significant.At the*α* = 0.01 level, we reject the null hypothesis and concludethe data are not statisticallysignificant. At the *α* = 0.01level, we fail to reject the null hypothesis and conclude the dataare statistically significant.At the *α* = 0.01 level, wefail to reject the null hypothesis and conclude the data are notstatistically significant.

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

There is sufficient evidence at the 0.01 level to conclude thatthe average hail damage to wheat crops in the county in Coloradodiffers from the national average.There is insufficient evidence atthe 0.01 level to conclude that the average hail damage to wheatcrops in the county in Colorado differs from the nationalaverage.

Answer:

(a) The level of significance is

Hypothesis :Vs

(b) The sampling distribution is

the standard normal, since we assume that *x* has anormal distribution with known *σ*.

The Z value of the sample test statistic is ,

(c) The p-value is ,

p-value=

; From standard normal distribution table

(d) Decision ; Here , p-value = 0.3370 >

Therefore , fail to reject Ho and conclude that the data are notstatistically significant

(e) Conclusion :

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