Question

1. An investment website can tell what devices are used to access the site. The site managers wonder whether they should enhance the facilities for trading via smartphones so they want to estimate the proportion of users who access the site using smartphones. They draw a random sample of 200 investors from their customers. Suppose that the true proportion of smartphone users is 36%.
   1. What would you expect the shape of the sampling distribution for the sample proportion to be?
      
   2. What should the mean of this sampling distribution be?
      
   3. If the sample size was increased to 500, would either of your answers to parts (a) or (b) change? Explain.
      
   4. What would the standard deviation of the sampling distribution of the proportion of smartphone users be?
      
   5. What is the probability that the sample proportion of smartphone users is above 0.36?
      
   6. What is the probability that the sample proportion of smartphone users is between 0.3 and 0.4?
      
   7. What is the probability that the sample proportion of smartphone users is below 0.28?

Answer 1

n = 200

p = 0.36

A.

Since np = 72 and n(1-p) = 128 10

Normal distribution

B.

Mean, = p = 0.36

C.

If n is increased to 500, then

np = 180 and n(1-p) = 320 10

Thus answer to part A won't' change.

Since the mean of the sampling distribution is independent of n, thus answer to part B also won't change.

D.

=

= 0.034

E.

P(>0.36) = P(Z>(0.36-0.36)/0.034)

= P(Z>0)

= 0.5

F.

P(0.3<0.4) = P((0.3-0.36)/0.034 < Z < (0.4-0.36)/0.034)

= P(-1.7647<Z<1.1765)

= 0.8415

G.

P(<0.28) = P(Z>(0.28-0.36)/0.034)

= P(Z>-2.353)

= 0.0093

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