SOCR Courses 2012 2013 Stat13 1 Lab4

Stats 13.1 - Laboratory Activity 4

Normal Probability Distribution Activity

Note: If at the end of this lab you feel like you could use some more examples, see: SOCR EduMaterials Activities Normal Probability examples

For the purposes of this lab, we are looking at \( N(\mu, \sigma) \).

Description: You can access the SOCR distribution applets here.

Exercise 1

Use SOCR to graph and print the distribution of \( X \sim N(20, 3) \). Show on the graph the following points\[\mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma \] by drawing hashmarks of where they are on the printout. How many standard deviations from the mean is the value \( x=27.5 \)?

Exercise 2

Graph the distribution of \( X \sim N(40, 10)\).

1. Find \(P(X > 49)\).
2. Find \(P(X < 22)\).
3. Find \(P(12 < X < 37)\). Submit a printout.
4. Use the mouse or the left cut off or right cut off points to find the \(8^{th}, 20^{th}, 45^{th}, 55^{th}, 70^{th}, 95^{th} \) percentiles. After you find these percentiles submit a printout for the \(95^{th}\) percentile.
5. Make sure you know how to answer the above questions using the \(z\) score \(z=\frac{x-\mu}{\sigma}\) and your \(z\) table from the handout! You do not need to submit anything here.

Exercise 3

The lifetime of tires of brand \(A\) follows the normal distribution with mean 40,000 miles and standard deviation 4,000 miles.

1. Use \(SOCR\) to find the probability that a tire will last between 40,000 and 46,000 miles.
2. Given that a tire will last more than 46,000 miles, what is the probability that it will last more than 50,000 miles? Explain how you get the answer.
3. Given that a tire will last more than 46,000 miles, what is the probability that it will last less than 50,000 miles? Submit a printout and explain how you get the answer.

Exercise 4

1. The probability that a student is admitted in the Math Department Major at a college is \(45 \%\). Suppose that this year 100 students will apply for admission into the Math major.
2. What is the distribution of the number of students admitted? Use SOCR to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
3. Write an expression for the exact probability that among the 100 students at least 55 will be admitted.
4. Use the appropriate applet in SOCR to compute the exact probability of part (3).
5. Use the normal distribution applet in SOCR to approximate the probability of part (3) (do not forget the continuity correction). What is the error of the approximation (i.e., how much does this differ from part (4))?

Below you can see the distribution of a normal random variable \( X \) with \( \mu=50, \sigma=5 \). In this graph you can also see the probability that \( X \) is between 53 and 60.

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• SOCR Home page: http://www.socr.ucla.edu

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