6.) Confidence Intervals and Sample Size A study of stress on the campus of the
ID: 3355249 • Letter: 6
Question
6.) Confidence Intervals and Sample Size A study of stress on the campus of the University of Illinois reported a mean stress level of 78 (on a 0 to 100 scale with a higher score indicating more stress) with a population standard deviation of 20. Calculate a 95 percent confidence interval for the following sample sizes, assume a normal distribution for the underlying population. You can find tables in A9 for the t distribution and A7 for the normal (Be aware: the table for the t is read as follows, the rows are the different degrees of freedom and the columns are each the probability of the random variable being less than or equal to the corresponding entry. Each entry is therefore the value for the given degree of freedom, that the probability given by the column that the random variable is less than or equal to that number. The normal table is read by setting the units and tenths digit using the row and the hundreths digit as the column, the entry is the probability that the normal random variable is less than that number): a.) 9 b.) 25 c.) 81 d.) 100 e.) How would you categorize the relationship between the confidence inter- val and the sample size? How would you explain this relationship?Explanation / Answer
Given the values as
Mean = 78
SD = 20
CI = 95% , so alpha = 0.05 , i.e 1-0.95 = 0.05
a)
N = 9
so degree of freedom
df = n-1 = 8
we know that the CI is given as
mean +- t*sd/sqrt(n)
here t = 2.306 , so putting the values we get
78 +- 2.306*20/sqrt(8) , we get the interval as
61.69 and 94.30
Following the same process and looking for the corresponding T values
b)
df = 25-1 = 24
mean +- t*sd/sqrt(n)
here t = 2.06 , so putting the values we get
78 +- 2.06*20/sqrt(25) , we get the interval as
69.76 and 86.24
c)
df = 81-1 = 80
mean +- t*sd/sqrt(n)
here t = 1.99 , so putting the values we get
78 +- 1.99*20/sqrt(81) , we get the interval as
73.57 and 82.4
d)
df = 100-1 = 99
mean +- t*sd/sqrt(n)
here t = 1.98 , so putting the values we get
78 +- 1.98*20/sqrt(100) , we get the interval as
74.04 and 81.96
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