Question

1. In this problem, assume that the distribution of differences is approximately normal. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more "conservative" answer.

Suppose that at five weather stations on Trail Ridge Road in Rocky Mountain National Park, the peak wind gusts (in miles per hour) for January and April are recorded below.

Wilderness District	1	2	3	4	5
January	139	120	126	64	78
April	101	110	108	88	61

Does this information indicate that the peak wind gusts are higher in January than in April? Use α = 0.01. Solve the problem using the critical region method of testing. (Let d = January − April. Round your answers to three decimal places.)

test statistic	=
critical value	=

Interpret your conclusion in the context of the application.

Reject the null hypothesis, there is sufficient evidence to claim average peak wind gusts are higher in January.

Fail to reject the null hypothesis, there is sufficient evidence to claim average peak wind gusts are higher in January.    

Fail to reject the null hypothesis, there is insufficient evidence to claim average peak wind gusts are higher in January.

Reject the null hypothesis, there is insufficient evidence to claim average peak wind gusts are higher in January.

Compare your conclusion with the conclusion obtained by using the P-value method. Are they the same?

We reject the null hypothesis using the critical region method, but fail to reject using the P-value method.

We reject the null hypothesis using the P-value method, but fail to reject using the critical region method.    

The conclusions obtained by using both methods are the same.

2.

Gentle Ben is a Morgan horse at a Colorado dude ranch. Over the past 8 weeks, a veterinarian took the following glucose readings from this horse (in mg/100 ml).

91

86

81

107

99

108

86

88

The sample mean is x ≈ 93.3. Let x be a random variable representing glucose readings taken from Gentle Ben. We may assume that x has a normal distribution, and we know from past experience that σ = 12.5. The mean glucose level for horses should be μ = 85 mg/100 ml.† Do these data indicate that Gentle Ben has an overall average glucose level higher than 85? Use α = 0.05.

(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 > 85; H₁:  μ = 85; right-tailed

H₀: μ = 85; H₁:  μ > 85; right-tailed    

H₀: μ = 85; H₁:  μ ≠ 85; two-tailed

H₀: μ = 85; H₁:  μ < 85; 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 unknown σ.

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 known σ.

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

What is the 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 = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.  

   At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 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 0.05 level to conclude that Gentle Ben's glucose is higher than 85 mg/100 ml.

There is insufficient evidence at the 0.05 level to conclude that Gentle Ben's glucose is higher than 85 mg/100 ml.     

Answer 1