Question

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​ "smart phones", so they want to estimate the proportion of users who access the site that way​ (even if they also use their computers​ sometimes). They draw a random sample of

200200

investors from their customers. Suppose that the true proportion of smart phone users is

3737​%.

​a) What would the standard deviation of the sampling distribution of the proportion of the smart phone users​ be?

. 034.034

​(Round to three decimal places as​ needed.)

​b) What is the probability that the sample proportion of smart phone users is greater than

0.370.37​?

. 5.5

​(Round to three decimal places as​ needed.)

​c) What is the probability that the sample proportion is between

0.320.32

and

0.420.42​?

. 858.858

​(Round to three decimal places as​ needed.)

​d) What is the probability that the sample proportion is less than

0.300.30​?

. 02.02

​(Round to three decimal places as​ needed.)

​e) What is the probability that the sample proportion is greater than

0.440.44​?

______

​(Round to three decimal places as​ needed.)

Answer 1

n=200 is the sample size

p=0.37 is the true proportion of smart phone users.

a) the standard deviation (also called the standard error) of the sampling distribution of the proportion of the smart phone users​ is

ans: the standard deviation (also called the standard error) of the sampling distribution of the proportion of the smart phone users​ is 0.034

Let indicate the sample proportion of smartphone users for a sample of size, n=200.

We can see that both

​

are greater than 5. That means using the central limit theorem, we can say that is normally distributed with mean and standard deviation

b) The probability that the sample proportion of smart phone users is greater than 0.37​ is

ans: The probability that the sample proportion of smart phone users is greater than 0.37​ is 0.50

c) The probability that the sample proportion is between 0.32 and 0.42 is

ans: The probability that the sample proportion is between 0.32 and 0.42 is 0.858

​d) The probability that the sample proportion is less than 0.30​ is

ans: The probability that the sample proportion is less than 0.30​ is 0.020

​e) The probability that the sample proportion is greater than 0.44​ is

ans:The probability that the sample proportion is greater than 0.44​ is 0.020