Question

1. The standardized IQ test is described as a normal distribution with 100 as the mean score and a 15-point standard deviation.

a. What is the Z-score for a score of 150?

b. What percentage of scores are above 150?

c. What percentage of scores fall between 85 and 150?

d. What does it mean to score in the 95th percentile?

e. What is the score that corresponds to being in the 95th percentile?

2. A friend wants to learn about the average weekly take-home pay for an Uber driver in Arkansas. After asking Uber to tell her this, they denied her request saying if she wanted to find out she would need to ask people on her own. With the help of a local ride-sharing driver organization, she obtained what she believes to be a list of all Uber drivers in the state of Arkansas. She asks you for help in figuring out what to do next.

a. What is the population and parameter of interest?

b. What is the sample statistic of interest?

3. After looking at the list you find out that there are 40,000 drivers listed. She doesn’t have money to contact 40,000 drivers in her research budget. Instead, she can talk to about 1,000. In your email reply to her, you tell her that it shouldn’t be a problem that she can only talk to about 1,000 drivers, assuming you select them the right way  

a. What is the “right way” to select the drivers? Make sure to explain what the “right way” is, not just provide the term.

b. Why do you tell her that it’s not a problem if you select them the right way? That is, what explanation do you give to her to explain that she can learn what she wants with the small sample? Your explanation should include the basics of probability sampling, sampling distributions, and the central limit theorem.

4. Agreeing to move forward with the research, she asks you to explain the different ways of developing a sample. She also mentioned she was interested in comparing the differences in earnings between women who are mothers, women who are not, and men.

a. How would you describe simple random sampling to her, and how would you construct a simple random sample from these data?

b. Describe systematic sampling to her and describe how you might construct such a sample from these data.

c. Describe stratified random sampling to her, how it might be useful for comparing the earnings between women who are mothers, those who aren’t, and men, and how you might construct such a sample from these data.

5. If the population parameter is $400, with a standard deviation of $100. What percentage of your potential sample statistics from the population of size 1,000 will fall between a value of $393.68 and $406.32?

Answer 1

Solution:

(1) We are given that: The standardized IQ test is described as a normal distribution with 100 as the mean score and a 15-point standard deviation.

That is: Mean =    and Standard Deviation =

Part a) What is the Z-score for a score of 150?

Part b) What percentage of scores are above 150?

That is: P( X > 150) = ...?

P( X > 150) = P( Z > 3.33)

P( X > 150) = 1 - P( Z < 3.33)

Look in z table for z = 3.3 and 0.03 and find area.

P( Z < 3.33 ) = 0.9996

Thus

P( X > 150) = 1 - P( Z < 3.33)

P( X > 150) = 1 - 0.9996

P( X > 150) = 0.0004

Part c) What percentage of scores fall between 85 and 150?

P( 85 < X < 150) = ....?

P( 85 < X < 150) = P( X < 150) - P( X < 85)

We have P( X < 150) = P(Z < 3.33 ) = 0.9996

Now find P( X < 85) .

Find z score:

Look in z table for z = -1.0 and 0.00 and find area.

P( Z < -1.00) = 0.1587

Thus P( X < 85) = P( Z < -1.00) = 0.1587

Thus

P( 85 < X < 150) = P( X < 150) - P( X < 85)

P( 85 < X < 150) = 0.9996 - 0.1587

P( 85 < X < 150) = 0.8409

Part d) What does it mean to score in the 95th percentile?

95th percentile means 95% of the IQ scores are below 95th percentile and 5% scores are above 95th percentile.

Part e) What is the score that corresponds to being in the 95th percentile?

P(X < x) = 0.95

Find z score value for 0.9500 area or its closest area.

Look in z table for Area = 0.9500 or its closest area and find corresponding z value.

Area 0.9500 is in between 0.9495 and 0.9505 and both the area are at same distance from 0.9500

Thus we look for both area and find both z values

Thus Area 0.9495 corresponds to 1.64 and 0.9505 corresponds to 1.65

Thus average of both z values is : ( 1.64+1.65) / 2 = 1.645

Thus z = 1.645

Thus we use following formula:

Thus the score that corresponds to being in the 95th percentile is x = 124.675