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Question $2$ , $5$ . $3$ . $30$ $\newline$ Points $\newline$ $x$ Points: $0$ of $1$ $\newline$ Find the $\newline$ $z$ -scores for which $\newline$ $15\%$ of the distribution's area lies between $\newline$ $-z$ and $\newline$ $z$ . $\newline$ Click to view pace lof the table Click to view page $2$ of the table. $\newline$ The $\newline$ $z$ -scores are $\newline$ $\square$ . $\newline$ (Use a comma to separate answers as needed. Round to two decimal places as needed.)

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Find the magnitude of a three-dimensional vector

Full solution

Q. Question

2

,

5

.

3

.

30

\newline

Points

\newline

x

Points:

0

of

1

\newline

Find the

\newline

z

-scores for which

\newline

15\%

of the distribution's area lies between

\newline

-z

and

\newline

z

.

\newline

Click to view pace lof the table Click to view page

2

of the table.

\newline

The

\newline

z

-scores are

\newline

\square

.

\newline

(Use a comma to separate answers as needed. Round to two decimal places as needed.)

Understand total distribution area: To find the z-scores for which $15\%$ of the distribution's area lies between $-z$ and $z$ , we need to understand that the total area under the standard normal distribution curve is $1$ (or $100\%$ ). Since we are looking for the area in the middle of the curve, we need to first determine the area in one tail to exclude.
Calculate excluded tail area: The area we want to exclude from each tail is half of $15\%$ , because we are looking for the area between $-z$ and $z$ , which is symmetrical around the mean. So, we calculate $15\% / 2 = 7.5\%$ for each tail.
Find z-score for $85$ % distribution: Now we need to find the z-score that corresponds to the area to the left of z that includes the middle $85\%$ of the distribution ( $100\% - 15\% = 85\%$ ). Since the distribution is symmetrical, we want the area to the left of z to be $85\% / 2 = 42.5\%$ plus the $7.5\%$ in the lower tail, which totals $50\%$ .
Lookup z-score for $0.9250$ area: Using a standard normal distribution table or a calculator with inverse normal distribution function, we look up the z-score that corresponds to an area of $0.5000 + 0.4250 = 0.9250$ to the left of $z$ .
Identify positive and negative z-scores: The z-score that corresponds to an area of $0.9250$ to the left is approximately $1.44$ . This is the positive z-score. Since the normal distribution is symmetrical, the negative z-score is $-1.44$ .
Final z-scores for $15$ % area: Therefore, the z-scores for which $15$ % of the distribution's area lies between $-z$ and $z$ are approximately $-1.44$ and $1.44$ .

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