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Mr. Emmer gave a test in his Chemistry class. The scores were normally distributed with a mean of 82 and a standard deviation of 4.

A student is randomly chosen. What is the probability that the student scores an 88 or below?

Use the formula for a [tex]$z$[/tex] score:

[tex]$ z=\frac{x-\mu}{\sigma} $[/tex]

where [tex][tex]$x$[/tex][/tex] is the given value, [tex]$\mu$[/tex] is the mean, and [tex]$\sigma$[/tex] is the standard deviation. Then refer to the chart on page 11 of the lesson to find the probability.

a. 0.9332

b. 0.8643

c. 0.9918

d. 0.6915



Answer :

GinnyAnswer
Let's solve this problem step-by-step.

### Step 1: Identify the given values
- Mean ([tex]\(\mu\)[/tex]): 82
- Standard deviation ([tex]\(\sigma\)[/tex]): 4
- Score ([tex]\(x\)[/tex]): 88

### Step 2: Calculate the z-score
The z-score formula is given by:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]

Plug in the given values:
[tex]\[ z = \frac{88 - 82}{4} \][/tex]
[tex]\[ z = \frac{6}{4} \][/tex]
[tex]\[ z = 1.5 \][/tex]

So, the z-score corresponding to the score of 88 is 1.5.

### Step 3: Find the probability using the z-score
Refer to the standard normal distribution table (the chart on page 11 of the lesson). The z-score of 1.5 corresponds to a cumulative probability.

From the standard normal distribution table, a z-score of 1.5 gives us a cumulative probability of approximately 0.9332.

### Step 4: Conclusion
The probability that a randomly chosen student scores 88 or below is approximately 0.9332.

Therefore, the correct answer is:
a. .9332

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