Lesson 2 Just ACT Normal Solidify Understanding

Learning Focus

Interpret standardized test scores that are distributed normally.

How can understanding normal distributions help us interpret test scores and other real-life data?

Open Up the Math: Launch, Explore, Discuss

One of the most common examples of a normal distribution is the distribution of scores on standardized tests like the ACT. In 2010, the mean score was $21$ and the standard deviation was $5.2$ (Source: National Center for Education Statistics).

Why does it make sense that scores are distributed normally on a standardized test?

1.

Use this information to sketch a normal distribution curve for this test.
Use technology to check your graph. Did you get the points of inflection in the right places? (Make adjustments, if necessary.)

2.

In What Is Normal, you learned the $68 – 95 – 99.7$ rule. Use the rule to answer the following problems:

a.

What percentage of students scored below $21$ ?

b.

About what percentage of students scored above $16$ ?

c.

About what percentage of students scored between $11$ and $26$ ?

3.

Your friend, Calvin, would like to go to a very selective college that only admits the top $1 %$ of all student applicants. Calvin has good grades and scored $33$ on the test. Do you think that Calvin’s ACT score gives him a good chance of being admitted? Explain your answer.

Pause and Reflect

4.

Many students like to eat microwave popcorn as they study for the ACT. Microwave popcorn producers assume that the time it takes for a kernel to pop is distributed normally with a mean of $120 seconds$ and a standard deviation of $13$ for a standard microwave oven. If you’re a devoted popcorn studier, you don’t want a lot of un-popped kernels, but you know that if you leave the bag in long enough to be sure that all the kernels are popped, some of the popcorn will burn. How much time would you recommend for microwaving the popcorn? Use a normal distribution curve and the features of a normal distribution to explain your answer.

Ready for More?

Here’s a challenge: Estimate the percent of test-takers who scored between $25$ and $30$ on the ACT.

Takeaways

Estimating population percentages from a normal distribution:

Lesson Summary

In this lesson, we sketched a normal distribution and used it to estimate population percentages. We found a process that helps to visualize the region included so that the $68 – 95 – 99.7 %$ rule can be used.

Retrieval

1.

You survey $100 students$ in your school and ask them if they would prefer to attend a pro-football game or a pro-basketball game. Only $25 students$ in the sample choose football. If you have $1,756 students$ total in your school, how many would you expect out of all the student body would choose football?

2.

Given: $\log \frac{100 a x^{2}}{7 b c}$ . Use the properties of logarithms to expand the expression as a sum or difference and/or constant multiple of logarithms. (Assume all variables are positive.)