Mathematics, 23.06.2020 19:01, lilspike420

IV. The College Board American College Testing Program reported a population mean SAT score of 960 (The New York Times Almanac). Assume that the population standard deviation is 100. 1.What is the probability that a random sample of 75 students will provide a sample mean SAT score within 10 of the population mean? 2. What is the probability that a random sample of 75 students will provide a sample mean SAT score within 20 of the population mean?

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Answers

1. 61.56% probability that a random sample of 75 students will provide a sample mean SAT score within 10 of the population mean

2. 91.64% probability that a random sample of 75 students will provide a sample mean SAT score within 20 of the population mean.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

When the distribution is normal, we use the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this question, we have that:

$\mu = 960, \sigma = 100, n = 75, s = \frac{100}{\sqrt{75}} = 11.547$

1.What is the probability that a random sample of 75 students will provide a sample mean SAT score within 10 of the population mean?

This is the pvalue of Z when X = 960 + 10 = 970 subtracted by the pvalue of Z when X = 960 - 10 = 950. So

X = 970

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{970 - 960}{11.547}$

Z = 0.87

Z = 0.87 has a pvalue of 0.8078

X = 950

$Z = \frac{X - \mu}{s}$

$Z = \frac{950 - 960}{11.547}$

Z = -0.87

Z = -0.87 has a pvalue of 0.1922

0.8078 - 0.1922 = 0.6156

61.56% probability that a random sample of 75 students will provide a sample mean SAT score within 10 of the population mean.

2. What is the probability that a random sample of 75 students will provide a sample mean SAT score within 20 of the population mean?

This is the pvalue of Z when X = 960 + 20 = 980 subtracted by the pvalue of Z when X = 960 - 20 = 940. So

X = 980

$Z = \frac{X - \mu}{s}$

$Z = \frac{980 - 960}{11.547}$

Z = 1.73

Z = 1.73 has a pvalue of 0.9582

X = 940

$Z = \frac{X - \mu}{s}$

$Z = \frac{940 - 960}{11.547}$

Z = -1.73

Z = -1.73 has a pvalue of 0.0418

0.9582 - 0.0418 = 0.9164

91.64% probability that a random sample of 75 students will provide a sample mean SAT score within 20 of the population mean.

answered: Guest

answer: a) (0,0), (1,1), (2,8), and (3,27).

step-by-step explanation:

0^3=0

1^3=1

2^3=8

3^3=27

in effect, the exponential equation for this would me y=3^x. each x-coordinate cubed is giving the y coordinate, so a is your answer : )

Pls i’m which set of ordered pairs could be generated by an exponential function a. (0,0), (1,,8),

answered: Guest

The answers are 20,30,80,180,960,890,1230,4560. on the first row
Can someone me with this! it’s for my little sister and she’s not sure what she has to do. asap

Can someone me with this! it’s for my little sister and she’s not sure what she has to do. asap

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IV. The College Board American College Testing Program reported a population mean SAT score of 960 (...