Question

learning goal
i can determine when a data set warrants a normal distribution. i
can determine the mean and standard deviation of a data set and
fit it to a normal distribution. i can estimate population
percentages based on mean, standard deviation, and distribution.
i can estimate the areas under the normal curve using calculators,
spreadsheets, and tables.
lesson 23.2 checkpoint
□ once you have completed the above problems and checked your solutions, complete the lesson checkpoint
below.
□ complete the lesson reflection above by circling your current understanding of the learning goal.
use the standard normal table below to answer question 2.
standard normal table ( z _ { x } = \frac { x - mu } { sigma } )

1. a college - entrance exam is designed so that scores are normally distributed with a mean of 500 and a

standard deviation of 100. what percent of exam scores are between 400 and 600?

1. suppose the scores on a test given to all juniors in a school district are normally distributed with a mean of

74 and a standard deviation of 8. using the standard normal table above, find the percent of juniors whose
score is less than 86.

Explanation:

Step1: Calculate z - scores for problem 1

For \(x = 400\), \(z_1=\frac{400 - 500}{100}=- 1\). For \(x = 600\), \(z_2=\frac{600 - 500}{100}=1\).

Step2: Find probabilities for problem 1

From the standard - normal table, \(P(Z\lt - 1)=0.1587\) and \(P(Z\lt1)=0.8413\). Then \(P(-1\lt Z\lt1)=P(Z\lt1)-P(Z\lt - 1)=0.8413 - 0.1587 = 0.6826\).

Step3: Calculate z - score for problem 2

For \(x = 86\), \(z=\frac{86 - 74}{8}=\frac{12}{8}=1.5\).

Step4: Find probability for problem 2

From the standard - normal table, when \(z = 1.5\), \(P(Z\lt1.5)\). Looking at the table: for \(z = 1.5\), the value is \(0.9332\).

Answer:

1. \(68.26\%\)
2. \(93.32\%\)