Scores on the SAT form a normal distribution with mu=500 and sigma=100. a) What is the minimum score necessary to be in the top 15% of the SAT distribution? Inviom

Calculate z-score: Step 1: Determine the z-score for the top 15% of the distribution. To find the z-score that corresponds to the top 15%, we look up or use a calculator for the percentile value. The z-score for the 85th percentile (since 100% - 15% = 85%) is approximately 1.036. Calculation: z-score for 85th percentile ≈ 1.036Use z-score formula: Step 2: Use the z-score formula to find the corresponding SAT score. We know the mean (μ) is 500 and the standard deviation (σ) is 100. Using the z-score formula: Z = (X - μ) / σ 1.036 = (X - 500) / 100 Calculation: X - 500 = 103.6Find minimum SAT score: Step 3: Solve for X to find the minimum SAT score. X = 103.6 + 500 Calculation: X = 603.6

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Scores on the SAT form a normal distribution with
mu=500 and
sigma=100.
a) What is the minimum score necessary to be in the top
15% of the SAT distribution? Inviom

$3$ . Scores on the SAT form a normal distribution with $\mu=500$ and $\sigma=100$ . $\newline$ a) What is the minimum score necessary to be in the top $15 \%$ of the SAT distribution?

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3

. Scores on the SAT form a normal distribution with

\mu=500

and

\sigma=100

\newline

a) What is the minimum score necessary to be in the top

15 \%

of the SAT distribution?

Calculate z-score: Step $1$ : Determine the z-score for the top $15$ \% of the distribution. $\newline$ To find the z-score that corresponds to the top $15$ \%, we look up or use a calculator for the percentile value. The z-score for the $85$ th percentile (since $100\% - 15\% = 85\%$ ) is approximately $1.036$ . $\newline$ Calculation: z-score for $85$ th percentile $\approx 1.036$
Use z-score formula: Step $2$ : Use the z-score formula to find the corresponding SAT score. $\newline$ We know the mean ( $\mu$ ) is $500$ and the standard deviation ( $\sigma$ ) is $100$ . Using the z-score formula: $\newline$ $Z = \frac{X - \mu}{\sigma}$ $\newline$ $1.036 = \frac{X - 500}{100}$ $\newline$ Calculation: $X - 500 = 103.6$
Find minimum SAT score: Step $3$ : Solve for $X$ to find the minimum SAT score. $X = 103.6 + 500$ Calculation: $X = 603.6$