Math  /  Data & Statistics

QuestionHomework \# 4: 7(1,2,3,4)8(2,3,4)7(1,2,3,4) 8(2,3,4) Question 7 of 40 (1 point) | Question Attempt: 1 of 3 1\checkmark 1 ×2\times 2 3\checkmark 3 ×4\times 4 5\equiv 5 6 7 8 9
A normal distribution has mean μ=61\mu=61 and standard deviation σ=20\sigma=20. Find and interpret the zz-score for x=63x=63.
The zz-score for x=63x=63 is \square . So 63 is \square standard deviations (Choose one) the mean μ=61\mu=61.

Studdy Solution

STEP 1

What is this asking? We're given a normal distribution with a **mean** of 6161 and a **standard deviation** of 2020, and we need to find and interpret the **z-score** for a value of 6363. Watch out! Don't mix up the mean (μ\mu) and the standard deviation (σ\sigma).
Also, remember that the z-score tells us how many standard deviations a value is away from the mean.

STEP 2

1. Calculate the z-score.
2. Interpret the z-score.

STEP 3

The **z-score** formula is given by: z=xμσ z = \frac{x - \mu}{\sigma} where xx is the **data point**, μ\mu is the **population mean**, and σ\sigma is the **population standard deviation**.

STEP 4

In our case, x=63x = \mathbf{63}, μ=61\mu = \mathbf{61}, and σ=20\sigma = \mathbf{20}.
Let's plug these values into the formula: z=636120 z = \frac{63 - 61}{20}

STEP 5

Now, we **subtract** in the numerator: z=220 z = \frac{2}{20} Then, we **divide** to get our z-score: z=220=12102=11022=1101=0.1 z = \frac{2}{20} = \frac{1 \cdot 2}{10 \cdot 2} = \frac{1}{10} \cdot \frac{2}{2} = \frac{1}{10} \cdot 1 = \mathbf{0.1}

STEP 6

Our calculated z-score is 0.1\mathbf{0.1}.
This means that the value 6363 is 0.1\mathbf{0.1} **standard deviations** *above* the **mean** of 6161.
It's *above* because the z-score is *positive*.

STEP 7

The z-score for x=63x = 63 is 0.1\mathbf{0.1}.
So 6363 is 0.1\mathbf{0.1} standard deviations *above* the mean μ=61\mu = 61.

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