Respuesta :
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.
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
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:
[tex]\mu = 960, \sigma = 100, n = 75, s = \frac{100}{\sqrt{75}} = 11.547[/tex]
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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{970 - 960}{11.547}[/tex]
[tex]Z = 0.87[/tex]
[tex]Z = 0.87[/tex] has a pvalue of 0.8078
X = 950
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{950 - 960}{11.547}[/tex]
[tex]Z = -0.87[/tex]
[tex]Z = -0.87[/tex] 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
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{980 - 960}{11.547}[/tex]
[tex]Z = 1.73[/tex]
[tex]Z = 1.73[/tex] has a pvalue of 0.9582
X = 940
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{940 - 960}{11.547}[/tex]
[tex]Z = -1.73[/tex]
[tex]Z = -1.73[/tex] 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.
