Question

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

90

82

105

100

112

82

91

The sample mean is x ≈ 94.1. 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; left-tailed

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

H₀: μ = 85; H₁:  μ > 85; right-tailed

H₀: μ = 85; H₁:  μ ≠ 85; two-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 Student's t, since n is large with unknown 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 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 = 0.05 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

Given that,
population mean(u)=85
standard deviation, σ =12.5
sample mean, x =94.1
number (n)=8
null, Ho: μ=85
alternate, H1: μ>85
level of significance, α = 0.05
from standard normal table,right tailed z α/2 =1.645
since our test is right-tailed
reject Ho, if zo > 1.645
we use test statistic (z) = x-u/(s.d/sqrt(n))
zo = 94.1-85/(12.5/sqrt(8)
zo = 2.059
| zo | = 2.059
critical value
the value of |z α| at los 5% is 1.645
we got |zo| =2.059 & | z α | = 1.645
make decision
hence value of | zo | > | z α| and here we reject Ho
p-value : right tail - ha : ( p > 2.059 ) = 0.02
hence value of p0.05 > 0.02, here we reject Ho
ANSWERS
---------------
a.
level of significance =0.05
null, Ho: μ=85
alternate, H1: μ>85
b.
The standard normal, since we assume that x has a normal distribution with known σ.
test statistic: 2.059
critical value: 1.645
decision: reject Ho
c.
p-value: 0.02
d.
At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
e.
we have enough evidence to support the claim that Gentle Ben's glucose is higher than 85 mg/100 ml.