1) A) The label of a certain brand of 1-ounce chocolate chip cookies states that
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Question
1) A) The label of a certain brand of 1-ounce chocolate chip cookies states that the cookies contain 100 calories. To make sure that the claim is not exceeded, a quality engineer routinely samples their cookies and checks the calorie content. A random sample of 14 1-ounce chocolate chip cookies resulted in the following calorie amounts:
90 115 140 130 115 145 150
90 110 130 125 110 100 135
Assume that is known to equal 24.12 and the calorie amounts are Normally distributed.
STATE:
At the = 0.05 level, is there sufficient evidence to conclude that the average calorie content of 1-ounce chocolate chip cookies produced by this brand is greater than 100 calories?
PLAN: Perform a test of significance for a population mean ( known). Set = 0.05.
What are the appropriate hypotheses for this procedure?
a. H0: = 100 calories and Ha: > 100 calories
b. H0: x = 100 calories and Ha: x > 100 calories
c. H0: = 100 calories and Ha: 100 calories
d. H0: x = 100 calories and Ha: x 100 calories
e. H0: = 100 calories and Ha: < 100 calories
f. H0: x = 100 calories and Ha: x < 100 calories
B)What is the parameter of interest in this study?
a. The true mean calorie amount of all 1-ounce chocolate chip cookies produced by this brand.
b. The true mean calorie amount of the sample of 14 1-ounce chocolate chip cookies produced by this brand.
c. The mean calorie amount of all 1-ounce chocolate chip cookies produced by all brands.
d. The true mean number of chocolate chips in all 1-ounce chocolate chip cookies produced by all brands.
C) In the SOLVE step of the procedure we must first check the conditions:
Randomization, Normality of the Sampling Distribution of x, and Known.
Are these conditions met? Why or why not?
a. Randomization is met because a random sample was taken from chocolate chip cookies made by this brand. Normality of the Sampling Distribution of x is met because it is given that calorie amounts are Normally distributed. Known is met as we have = 24.12.
b. Randomization is not met because a random sample was not taken from all chocolate chip cookies made by all brands. Normality of the Sampling Distribution of x is met because it is given that calorie amount is Normally distributed. Known is met as we have = 24.12.
c. Randomization is not met because a random sample was not taken from all chocolate chip cookies made by all brands. Normality of the Sampling Distribution of x is not met because the plot of the histogram of calorie amount is left-skewed. Known is met as we have = 24.12.
d. Randomization is met because a random sample was taken from all chocolate chip cookies made by all brands. Normality of the Sampling Distribution of x is not met because the plot of the histogram of calorie amount is left-skewed. Known is met as we have = 24.12.
D)The SOLVE step of the procedure yielded an x = 120.35714. What is the proper z-statistic for this test?
a. z = 3.16
b. z = 3.05
c. z = 2.87
d. z = -1.93
e. z = -3.16
E) As part of the SOLVE step, suppose the z-statistic was 3.22, what would the proper p-value be for this test?
a. 0.0006
b. 0.9994
c. 0.0012
d. < 0.0001
F)If the p-value for the test was 0.0042 (it's not) what would you conclude at the = 0.05 significance level, in context?
a. Since the p-value for this test (0.0042) was less than the 5% significance level, we have sufficient evidence to reject the null hypothesis and conclude that the true mean calorie amount of all 1-ounce chocolate chip cookies produced by this brand is greater than 100 calories.
b. Since the p-value for this test (0.0042) was less than the 5% significance level, we fail to reject the null hypothesis, so we do not have sufficient evidence to conclude that the true mean calorie amount of all 1-ounce chocolate chip cookies produced by this brand is greater than 100 calories.
c. Since the p-value for this test (0.0042) was greater than the 5% significance level, we have sufficient evidence to conclude that the true mean calorie amount of all 1-ounce chocolate chip cookies produced by this brand is greater than 100 calories.
d. Since the p-value for this test (0.0042) was less than the 5% significance level, we fail to reject the null hypothesis, we have sufficient evidence to conclude that the true mean calorie amount of all 1-ounce chocolate chip cookies produced by this brand is greater than 100 calories.
Explanation / Answer
A.
a. H0: = 100 calories and Ha: > 100 calories [ANSWER]
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B.
a. The true mean calorie amount of all 1-ounce chocolate chip cookies produced by this brand. [ANSWER]
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C.
a. Randomization is met because a random sample was taken from chocolate chip cookies made by this brand. Normality of the Sampling Distribution of x is met because it is given that calorie amounts are Normally distributed. Known is met as we have = 24.12. [ANSWER]
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D.
Formulating the null and alternative hypotheses,
Ho: u <= 100
Ha: u > 100
As we can see, this is a right tailed test.
Getting the test statistic, as
X = sample mean = 120.35714
uo = hypothesized mean = 100
n = sample size = 14
s = standard deviation = 24.12
Thus, z = (X - uo) * sqrt(n) / s = 3.157937117 = 3.16 [ANSWER, A]
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e.
If z = 3.22 for a right tailed test,
P = 0.0006 [ANSWER, A]
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f.
a. Since the p-value for this test (0.0042) was less than the 5% significance level, we have sufficient evidence to reject the null hypothesis and conclude that the true mean calorie amount of all 1-ounce chocolate chip cookies produced by this brand is greater than 100 calories. [ANSWER]
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