MONT 105Q – Mathematical Journeys Problem Set 4 – Using the Standard Normal Table due: March 30, 2016

Use the standard normal table to the following questions.

A) Let Z be a standard normal.

1) Find P (Z > .57).

2) Find P (Z < −1.3).

3) Find P (.37 < Z < 1.20).

4) Find P (−.55 < Z < 1.30).

5) Find c such that P (Z > c) = .01. (You’ll probably want to “interpolate.”)

6) How would your answer to 5) change if it said: Find c such that P (|Z| > c) = .01? (Note: |Z| > c says: the absolute value of Z is greater than c. That can happen if either Z > c or Z < −c.)

B) Y is normally distributed with mean 10 and SD 3. Find

1) P (Y < 9).

2) P (Y > 11.3).

3) P (7 < Y ≤ 16). How does this answer compare with the “rule of thumb” given by Wheelan in the graph on page 26 of Naked Statistics?)

C) For this question, you will need to combine what we learned before about the Central Limit Theorem with our new techniques based on the standard normal table. Let X be the maximal oxygen intake of a human on a treadmill where the measurements are made in milliliters of oxygen per minute per kilogram of weight. Assume that for a particular population of patients the population mean of X is μ = 54.030 and the population standard deviation is σ = 5.8. Let \overline{X} be the sample mean of a random sample of size n = 50 from this population.

1) Find P (52.761 ≤ \overline{X} ≤ 54.453).

2) (Without calculation, and explain): Would the answer to part 1) be larger or smaller if the sample size was n = 100?

3) Suppose 10 different random samples of size n = 50 are taken from this population. What is the probability that exactly 7 of them will have 52.761 ≤ \overline{X} ≤ 54.453?