The College Board reported the following mean scores for the...

The College Board reported the following mean scores for the three parts of the Scholas- tic Aptitude Test (SAT) (The World Almanac, 2009): $\begin{array}{ll}{\text { Critical Reading }} & {502} \\ {\text { Mathematics }} & {515} \\ {\text { Writing }} & {494}\end{array}$ Assume that the population standard deviation on each part of the test is $\sigma=100$ . a. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the Critical Reading part of the test? b. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 515 on the Mathematics part of the test? Compare this probability to the value computed in part (a). c. What is the probability that a random sample of 100 test takers will provide a sample mean test score within 10 of the population mean of 494 on the writing part of the test? Comment on the differences between this probability and the values computed in parts (a) and (b).

The College Board reported the following mean scores for the three parts of the Scholas-
tic Aptitude Test (SAT) (The World Almanac, 2009):
$\begin{array}{ll}{\text { Critical Reading }} & {502} \\ {\text { Mathematics }} & {515} \\ {\text { Writing }} & {494}\end{array}$
Assume that the population standard deviation on each part of the test is $\sigma=100$ .
a. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the Critical Reading part of the test?
b. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 515 on the Mathematics part of the test? Compare this probability to the value computed in part (a).
c. What is the probability that a random sample of 100 test takers will provide a sample mean test score within 10 of the population mean of 494 on the writing part of the test? Comment on the differences between this probability and the values computed in parts (a) and (b).

Key Concepts

Sampling Distribution of the Sample Mean

This concept explains that when samples of a fixed size are drawn from a population, the distribution of their means tends to follow a specific pattern. Even if the population is not normally distributed, the distribution of the sample mean approximates normality for large sample sizes, which is crucial for making probability statements about the sample mean.

Central Limit Theorem

The Central Limit Theorem is a fundamental principle in statistics that states that the distribution of the sample mean will tend to be normally distributed as the sample size increases, regardless of the shape of the population distribution. This theorem justifies the use of normal probability models when estimating probabilities for sample means, particularly when sample sizes are sufficiently large.

Standard Error of the Mean

The Standard Error of the Mean quantifies the variability of the sample mean from one sample to another and is calculated by dividing the population standard deviation by the square root of the sample size. It serves as a measure of precision for the sample mean as an estimate of the population mean, which is key to understanding the margin within which sample means fall.

Z-Score Transformation

This concept involves converting a sample mean into a standardized value by subtracting the population mean and dividing by the Standard Error. The resulting Z-score represents the number of standard deviations the sample mean is away from the population mean and enables the use of the standard normal distribution to compute probabilities.

Probability Calculation Using the Standard Normal Distribution

Once the sample mean is standardized to a Z-score, probability calculations can be performed using the standard normal distribution. This process involves finding the area under the normal curve that corresponds to the desired interval, such as within a certain margin from the population mean, thus quantifying the likelihood of observing such a sample mean.

Transcript

00:01 All right, we're given some statistics about the population mean score on the critical reading, math, and writing sections of the sat.

00:09 And we're also given that the population standard deviation for all three of these categories is 100.

00:14 Now, for each of these parts, we're given a sample size, a section, and then we're supposed to find the probability that our point estimate for the mean will fall under a certain interval.

00:25 So for part a, our sample size is 90.

00:28 We're looking at the crit reading section and our error the error we're trying to find the probability of is plus or minus 10.

00:36 Really technically none of this writing and math and critical reading stuff.

00:42 The scores don't matter because we're only looking at the error.

00:46 So, yeah, keep that in mind.

00:49 Anyway, so let's start with our standard deviation of the sampling distribution.

00:59 I think it's to say that 90 is small enough compared to all the people who take the sat that we can just assume that our population size is relatively infinite.

01:14 So we're just going to use that formula.

01:18 So it's going to be 100 over the square root of 90.

01:23 This equals about 10 .54.

01:26 Now we're going to z score the upper and lower bounds.

01:30 So negative 10 gets z scored to approximately 0 .95.

01:40 On the lower, it's negative, it's positive 10, not negative, over 10 .54.

01:50 Okay, looking at our table, since by central limit theorem, this is big enough that we can just assume it's normal.

01:57 Probability on the lower end is 0 .1711.

02:02 Probability on the upper end is 0 .8 -89.

02:08 We find our probability by subtracting the upper limit minus the lower limit.

02:15 And this equals 0 .6758.

02:25 All right, part b, sample size is still 90.

02:29 Now we're looking at the math section, and we're still looking for an error of 10.

02:34 But you probably notice in part a, we never used this bit of information here.

02:39 Like, the population mean never came into it, so we don't need to worry about it.

02:43 In fact, we don't even need to worry about it here or here.

02:47 It's completely irrelevant to the question.

02:50 Anyway, so let's do this...

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