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d) A score that is $30$ points below the mean. $\newline$ $2$ . The Welcher Adult Intelligence Test Scale is composed of a number of subtests. On one subtest, the raw scores have a mean of $35$ and a standard deviation of $6$ . Assuming these raw scores form a normal distribution $\newline$ a) What number represents the $65^{\text{th}}$ percentile (what number separates the lower $65\%$ of the distribution)?

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Find probabilities using the normal distribution II

Full solution

Q. d) A score that is

30

points below the mean.

\newline

2

. The Welcher Adult Intelligence Test Scale is composed of a number of subtests. On one subtest, the raw scores have a mean of

35

and a standard deviation of

6

. Assuming these raw scores form a normal distribution

\newline

a) What number represents the

65^{\text{th}}

percentile (what number separates the lower

65\%

of the distribution)?

Understand Problem: Understand the problem and identify the formula to use. $\newline$ We need to find the $65$ th percentile of a normal distribution. The formula for any point in a normal distribution is: $\newline$ $X = \mu + Z \times \sigma$ $\newline$ where $\mu$ is the mean, $\sigma$ is the standard deviation, and $Z$ is the Z-score corresponding to the desired percentile.
Find Z-score: Find the Z-score for the $65$ th percentile. $\newline$ Using a Z-table, the Z-score that corresponds to the $65$ th percentile is approximately $0.385$ .
Calculate Score: Calculate the score corresponding to the $65$ th percentile. $\newline$ Plug the values into the formula: $\newline$ $X = 35 + 0.385 \times 6$ $\newline$ $X = 35 + 2.31$ $\newline$ $X = 37.31$

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