Answer:
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.
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
and standard deviation
, the zscore of a measure X is given by:
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
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
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:
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
By the Central Limit Theorem
has a pvalue of 0.8078
X = 950
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
has a pvalue of 0.9582
X = 940
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.