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d) A score that is 30 points below the mean. scores have a mean of 35 and a standard deviation of 6 . Assuming these raw scores form a normal distribution
a) What number represents the
65^("th ") percentile (what number separates the lower
65% of the distribution)?

d) A score that is $30$ points below the mean. 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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The Central Limit Theorem

Full solution

Q. d) A score that is

30

points below the mean. 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)?

Identify mean and deviation: Step $1$ : Identify the mean and standard deviation. $\newline$ Mean ( $\mu$ ) = $35$ $\newline$ Standard deviation ( $\sigma$ ) = $6$
Convert to z-score: Step $2$ : Convert the $65^{\text{th}}$ percentile to a z-score. $\newline$ Using the standard normal distribution table, the z-score corresponding to the $65^{\text{th}}$ percentile is approximately $0.385$ .
Calculate raw score: Step $3$ : Calculate the raw score corresponding to the $65$ th percentile. $\newline$ Raw Score = $\mu + z \times \sigma$ $\newline$ Raw Score = $35 + 0.385 \times 6$ $\newline$ Raw Score = $35 + 2.31$ $\newline$ Raw Score = $37.31$

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