Lesson 4 Wow, That’s Weird! Practice Understanding

Learning Focus

Compare normal distributions.

What methods can we find for using mean and standard deviation to compare normal distributions?

How can we decide if an event that seems weird is actually unusual or fairly likely?

Open Up the Math: Launch, Explore, Discuss

Each of the scenarios below is based upon normal distributions. Rank these scenarios from most unusual to most average. ( $1$ is the most unusual, $8$ is the most average.) In each case, explain your ranking.

1.

a.

The number of red loops in a box of Tutti-NoFrutti-Os is normally distributed with mean of $800$ loops and a standard deviation of $120$ . Tony bought a new box, opened it, and counted $1, 243$ red loops.

Rank: $\underset{―}{}$

Explanation:

b.

The weight of house cats is normally distributed with a mean of $10 pounds$ and standard deviation of $2.1 pounds$ . My cat, Big Boy, weighs $6 pounds$ .

Rank: $\underset{―}{}$

Explanation:

c.

The lifetime of a battery is normally distributed with a mean life of $40 hours$ and a standard deviation of $1.2 hours$ . I bought a battery and it stopped working after just $20 hours$ .

Rank: $\underset{―}{}$

Explanation:

d.

The amount that a human fingernail grows in a year is normally distributed with a mean length of $3.5 cm$ and a standard deviation of $0.63 cm$ . My neighbor’s thumbnail grew all year without breaking and it is $4.6 cm$ long.

Rank: $\underset{―}{}$

Explanation:

e.

My little brother was digging in the garden and found a giant earthworm that was $35 cm$ long. The length of earthworms is normally distributed with a mean length of $14 cm$ and a standard deviation of $5.3 cm$ .

Rank: $\underset{―}{}$

Explanation:

f.

The mean length of a human pregnancy is $268 days$ with a standard deviation of $16 days$ . My aunt just had a premature baby delivered after only $245 days$ .

Rank: $\underset{―}{}$

Explanation:

g.

IQ scores for young adults on a famous IQ test are distributed normally with a mean of $110$ and a standard deviation of $25$ . I’m pretty smart and my IQ is $135$ .

Rank: $\underset{―}{}$

Explanation:

h.

The army measured head sizes among male soldiers and found that the distribution is pretty close to normal with a mean of $22.8 inches$ and a standard deviation of $1.1 inches$ . Little Joe was almost too small to get into the army because his head size was only $20.6 inches$ .

Rank: $\underset{―}{}$

Explanation:

Ready for More?

Write your own weird situation like the ones given in the task with:

a.

A $z$ -score between $2$ and $3$ .

b.

A z-score between $0$ and $- 1$ .

c.

A $z$ -score between $- 2$ and $- 1$ .

Takeaways

Strategies for comparing events on normal distributions:

Lesson Summary

In this lesson, we used features of a normal distribution—including the symmetry, the mean, and the standard deviation—to determine how unusual a given event may be.

Retrieval

1.

Before teaching a lesson about boarding cats and dogs, Mrs. Jones decided to survey each of her classes to find out about her students’ animal preferences. Here is her data:

1st hour	2nd hour	3rd hour
C C C C C C C C C C C C C C C C C C C C C C C D D D D D D D D N N	C C C C C C C C C C C C D D D D D D D D D D D D D D D D D D D D N N N	C C C C D D D D D D D D D D D D D D D D D D D D D D D D D N N N N N

1st hour

2nd hour

3rd hour

C C C C C C C C C C C C

C C C C C C C C C C C

D D D D D D D D

N N

C C C C C C C C C C C C

D D D D D D D D D D

N N N

C C C C

D D D D D D D D D D D D D D D

D D D D D D D D D D

N N N N N

	1st	2nd	3rd	total
prefer cats
prefer dogs
no pets - no preference
total

2.

Solve for $x$ by applying properties of exponents and logarithms.

$\log (5 x^{2} + 3 x - 6) = \log (- x^{2} - 4 x - 1)$