Question

A population of values has a normal distribution with u = 44.6 and 0 = 51.1. You intend to draw a random sample of size n = 230. Find P93, which is the score separating the bottom 93% scores from the top 7% scores. P 93 (for single values) = Find P93, which is the mean separating the bottom 93% means from the top 7% means. P93 (for sample means) = Enter your answers as numbers accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

Solution:-

Given that,

mean = $\mu$ = 44.6

standard deviation = $\sigma$ = 51.1

P₉₃

Using standard normal table,

P(Z < z) = 93%

= P(Z < z) = 0.93  

= P(Z < 1.476) = 0.93

z = 1.476

1) Using z-score formula,

x = z * $\sigma$ + $\mu$

x = 1.476 * 51.1 + 44.6

x = 120.0

2) n = 230

$\mu$_{$\bar x$} = $\mu$ = 44.6

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 51.1/ $\sqrt$ 230 = 3.37

Using z-score formula  

$\bar x$ = z * $\sigma$ _{$\bar x$}+ $\mu$ _{$\bar x$}

$\bar x$ = 1.476 * 3.37 + 44.6

$\bar x$ = 49.6