Applied Stats HW #5 1. \"Are you smarter than a 5th grader?\" Suppose college fa

Question

Applied Stats HW #5 1. "Are you smarter than a 5th grader?" Suppose college faculty members are given 32 questions from the show, "Are You Smarter than a 5th grader?" Let X represent the number a faculty member gets correct, out of 32. Suppose the value of X varies among the population of faculty members according to the distribution below. Note that the population mean and standard deviation are 20.0 and 10.0, respectively. Ox Samping Distributions Me 2000 muddin 21 00 Part distribution with rosie 1000 shev 036 kuulasi- .080 Clear lower 3 Custom Sample Dete Sample Animated 1,000 10000 Distribution of Menna, H25 CFE normal a) Suppose 25 faculty members are selected at random and asked the 32 questions. How would the data (number correct) for these 25 be distributed? What is the mean, sd, and shape you would expect? b) Suppose thousands of different random samples of 25 faculty are selected and the mean is calculated for each sample. How would these means be distributed? Why? What is the mean, sd, and shape would you expect? c) Find the probability that the mean number of questions correct for a random sample of 25 faculty would be less than 15.9.

Explanation / Answer

a)
mean = 20
sd = 10/sqrt(25) = 2

The shape of data would be approximately normal.

b)
Means would be distributed normal.
mean = 20
sd = 10/sqrt(25) = 2

c)
P(X < 15.9)
= P(t < (15.9 - 20)/2)
= P(t < -2.05)
= 0.0257 (using standard normal t-table)

Here we used t-distribution because sample size is less than 30 though the population std.dev. is given.

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