Researchers have claimed that the average number of headaches during a semester
ID: 3159176 • Letter: R
Question
Researchers have claimed that the average number of headaches during a semester of Statistics is 14. Statistics professors dispute this claim vehemently. Statistic professors believe the average is much more than this. They randomly sample n = 40 students and find the sample mean is 16 and the sample standard deviation is 2.0.
(a)Define the parameter of interest to the statistic professors
(b) State the null and alternative hypotheses of interest to the professors.
Psychologists studying post-traumatic stress disorder (PTSD) will obtain a random sample of 64 Vietnam war POW survivors. Each of the 64 POW survivors will be given a test which measures level of PTSD.
(a) Identify the population of interest to the researchers.
(b) Identify the variable of interest to the researchers.
(c) Suppose the mean test score for all Vietnam war POW survivors is µ = 10 with a standard deviation given by = 2. Give the sampling distribution of X. Justify your answer.
(d) Find the probability the sample mean will exceed 10.5. (e) Interpret this probability.
(f) Suppose the mean test score, (µ), for all Vietnam war POW survivors is unknown. A 95% CI for µ is given by (10.82, 12.56). Interpret this interval.
(g) Is the mean score equal to 10?
Explanation / Answer
Researchers have claimed that the average number of headaches during a semester of Statistics is 14. Statistics professors dispute this claim vehemently. Statistic professors believe the average is much more than this. They randomly sample n = 40 students and find the sample mean is 16 and the sample standard deviation is 2.0.
(a)Define the parameter of interest to the statistic professors
Solution:
The parameter of interest to the statistic professors is number of headaches.
(b) State the null and alternative hypotheses of interest to the professors.
Solution:
The null and alternative hypothesis is given as below:
Null hypothesis: H0: The average number of headaches during a semester of Statistics is 14.
Alternative hypothesis: Ha: The average number of headaches during a semester of Statistics is more than 14.
Psychologists studying post-traumatic stress disorder (PTSD) will obtain a random sample of 64 Vietnam war POW survivors. Each of the 64 POW survivors will be given a test which measures level of PTSD.
(a) Identify the population of interest to the researchers.
Solution:
The population of interest to the researchers is the all Vietnam war POW survivors.
(b) Identify the variable of interest to the researchers.
Solution:
The variable of interest to the researchers is the post traumatic stress disorder in the Vietnam war POW survivors.
(c) Suppose the mean test score for all Vietnam war POW survivors is µ = 10 with a standard deviation given by = 2. Give the sampling distribution of X. Justify your answer.
Solution:
The mean for the sampling distribution is given as 10 and the standard deviation for the sampling distribution of X is given as /sqrt(n) = 2/sqrt(64) = 2/8 = ¼ = 0.25.
(d) Find the probability the sample mean will exceed 10.5.
Solution:
P(X>10.5) = 1 – P(X<10.5)
Z = (10.5 – 10) / 0.25 = 0.5/0.25 = 2
P(Z<2) = 0.97725
P(X>10.5) = 1 – P(X<10.5) = 1 – P(Z<2) = 1 – 0.97725 = 0.02275
Required probability = 0.02275
(e) Interpret this probability.
Solution:
The given probability is less than 0.05, so it is significant or improper.
(f) Suppose the mean test score, (µ), for all Vietnam war POW survivors is unknown. A 95% CI for µ is given by (10.82, 12.56). Interpret this interval.
Solution:
We are 95% confident that the average test score for all Vietnam war POW survivors is lies between the two values 10.82 and 12.56.
(g) Is the mean score equal to 10?
Solution:
No, the mean score could not be equal to 10 because it is not included in the given confidence interval.
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