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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Answer from: ehaynie

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.

Answer from: Quest

step-by-step explanation:

i believe its "c" because all of the other options include a but it could be d because there is a way of financing with cash at certain places but its uncommon.

Answer from: Quest

they disappear if your absent from to long. i found out that the hard way.

step-by-step explanation:

brainliest lol i just need one more brainliest.

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