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).

95

90

84

104

99

111

86

88

The sample mean is x ? 94.6. 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 ? = 12.5. The mean glucose level for horses should be ? = 85 mg/100 ml.† Do these data indicate that Gentle Ben has an overall average glucose level higher than 85? Use ? = 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: ? = 85; H₁:  ? ? 85; two-tailed

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

H₀: ? = 85; H₁:  ? < 85; left-tailed

H₀: ? > 85; H₁:  ? = 85; right-tailed

(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 ?.

The Student's t, since we assume that x has a normal distribution with known ?.     

The standard normal, since we assume that x has a normal distribution with unknown ?.

The standard normal, since we assume that x has a normal distribution with known ?.

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 ??

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

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

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

At the ? = 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

(a)

The level of significance is .

The test is right tailed test as the aim is to test whether the glucose level is higher than 85. The null and alternate hypotheses for this test are

(b)

Although the sample size is small, we have the assumption of normal distribution and the population standard deviation is known. So the standard normal distribution can be used as sampling distribution. The correct option is

"The standard normal, since we assume that x has a normal distribution with known ."

The test statistic for the standard normal distribution is given as

Substitute the values in the above test statistic.

The value of the sample test statistic is 2.17.

(c)

The P-value for the test is given as

The P-value for the test is 0.0150.

(d)

The P-value is 0.0150 which is lesser than the level of significance 0.05. So there is suffcient evidence to reject the null hypothesis at 5% level. The correct statement is

"At the level we reject the null hypothesi and conclude the data are statistically signfiicant"

(e)

As the null hypothesis is rejected, it is reasonable to conclude that Gentle Ben's glucose is higher than 85mg/100ml. The correct statement is

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