Question

12. Using a normal distribution and z score formula answer the following questions: a. Find the Z-score that cuts off the top 35% of the nornval curve b. Find the data value to the nearest whole number that cuts off the bottom 20% of the curve given that the mean is 75 and sample standard deviation is 5. c. Find the z scores that cut off the middle 50% of the normal curve. 13. This question is worth 4 points only. A statistics professor has surveyed 225 NCC students to determine the cost of their textbooks for the semester. He determined that the sample mean was $425 with a population standard deviation of $75 Determine the standard error of a 95% confidence interval Determine the 95% confidence interval (round to the nearest whole number) Determine the 99% confidence interval (round to the nearerst whole #t)

Answer 1

ANSWER:

12)

a)

Given

35% of the normal curve

P(Z < z) = 35%

P(Z < z) = 0.35

P(Z < -0.39) = 0.35

z = -0.39 (using Z - table)

b)

mean = $\small \mu$ =75

standard deviation = $\small \sigma$ = 5

P(Z < z) = 0.20

P(Z < -0.84) = 0.20

Therefore the cuttoff value here is computed as:

D -X

$\small z\sigma =X-\mu \Rightarrow X= \mu -z\sigma$

= $\small \mu$ - 0.84*$\small \sigma$

= 75 - 0.842*5

x = 70.79

c)

From standard normal tables, we have here:
P(Z < 0.67) = 0.7486 = 0.75

P(Z < -0.67) = 0.2514 = 0.25

P(Z < 0.67) - P(Z < -0.67) = 0.75 - 0.25 = 0.5

Therefore, due to symmetry, we have here:
P(-0.67 < Z < 0.67) = 0.5

Therefore -0.674, 0.674 are the required z scores.

..............................................................

13)

i)

n = 225

mean = $\small \overline{X}$ = 425

std.deviation = $\small \sigma$ =75

std.error = S.E =  $\small \frac{\sigma }{\sqrt{n}}$

$\small =\frac{75}{\sqrt{225}}$

S.E = 5

ii)

1 - $\small \alpha$ = 0.95

Z_{$\small \alpha$/2} = 1.96

$\small =\overline{X}\pm Z_{\alpha /2}\frac{\sigma }{\sqrt{n}}$

$\small =425\pm 1.96*5$

= 425 $\small \pm$ 9.8

(415.2 , 434.8)

iii)

1 - $\small \alpha$ = 0.99

Z_{$\small \alpha$/2} = 2.57

$\small =\overline{X}\pm Z_{\alpha /2}\frac{\sigma }{\sqrt{n}}$

$\small =425\pm 2.576*5$

= 425 $\small \pm$ 12.88

(412.12 , 437.88)

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