A recent survey conducted by Gallup (an analytic and advice firm) estimated that 50% of students in Massachusetts prefer learning full-time in person, while the remaining 50% prefer a full-time remote or a hybrid arrangement. Suppose we assume that, among USC students, the proportion of those who prefer learning full-time in person instruction is the same as that estimated by Gallup for Massachusetts. We interview a random sample of 70 USC students regarding their preferences over full-time in person versus remote/hybrid instruction. c) Under our assumption that the preferences of USC students are the same as those estimated by Gallup for Massachusetts, using the approximation based on the Normal distribution, what is the sampling distribution of the proportion of interviewed students who prefer full-time in person instruction? d) Under our assumption that the preferences of UCI students are the same as those estimated by Gallup for Massachusetts, using the approximation based on the Normal distribution that you ob- tained in part (c), (i) what is the probability that less than 28 of the 70 interviewed students declare to prefer full-time in person instruction over remote/hybrid arrangements? (ii) what is the probability that up to 28 of the 70 interviewed students declare to prefer full-time in person instruction over remote/hybrid arrangements?
A recent survey conducted by Gallup (an analytic and advice firm) estimated that 50% of students in Massachusetts prefer learning full-time in person, while the remaining 50% prefer a full-time remote or a hybrid arrangement. Suppose we assume that, among USC students, the proportion of those who prefer learning full-time in person instruction is the same as that estimated by Gallup for Massachusetts. We interview a random sample of 70 USC students regarding their preferences over full-time in person versus remote/hybrid instruction.
c) Under our assumption that the preferences of USC students are the same as those estimated by Gallup for Massachusetts, using the approximation based on the
d) Under our assumption that the preferences of UCI students are the same as those estimated by Gallup for Massachusetts, using the approximation based on the Normal distribution that you ob- tained in part (c), (i) what is the
e) Compare the exact probabilities computed in part (b) with the approximate probabilities computed in part (d): sort them from lowest to greatest. Would you say that the Normal approximation is a good approximation?
f) Compare the mean of the exact probabilities that you computed in part (b) with the approximate probabilities computed in part (d). You should find that they are very close. Why do you think that this is the case? (Hint: this has something to do with the fact that your answers to d(i) and d(ii) should actually be the same, while the answers to b(i) and b(ii) should not).
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