The scores of all applicants taking an aptitude test...

The scores of all applicants taking an aptitude test required by a law school have a normal distribution with a mean of 420 and a standard deviation of 100 . A random sample of 25 scores is taken. a. Find the probability that the sample mean score is higher than 450 . b. Find the probability that the sample mean score is between 400 and 450 . c. The probability is 0.10 that the sample mean score is higher than what number? d. The probability is 0.10 that the sample mean score is lower than what number? e. The probability is 0.05 that the sample standard deviation of the scores is higher than what number? f. The probability is 0.05 that the sample standard deviation of the scores is lower than what number? g. If a sample of 50 test scores had been taken, would the probability of a sample mean score higher than 450 be smaller than, larger than, or the same as the correct answer to part (a)? It is not necessary to do the detailed calculations here. Sketch a graph to illustrate your reasoning.

The scores of all applicants taking an aptitude test required by a law school have a normal distribution with a mean of 420 and a standard deviation of 100 . A random sample of 25 scores is taken.
a. Find the probability that the sample mean score is higher than 450 .
b. Find the probability that the sample mean score is between 400 and 450 .
c. The probability is 0.10 that the sample mean score is higher than what number?
d. The probability is 0.10 that the sample mean score is lower than what number?
e. The probability is 0.05 that the sample standard deviation of the scores is higher than what number?
f. The probability is 0.05 that the sample standard deviation of the scores is lower than what number?
g. If a sample of 50 test scores had been taken, would the probability of a sample mean score higher than
450 be smaller than, larger than, or the same as the correct answer to part (a)? It is not necessary to do the detailed calculations here. Sketch a graph to illustrate your reasoning.

Key Concepts

Chi-Squared Distribution

This distribution is used when analyzing the variability of sample variances from a normally distributed population. Specifically, when sampling from a normal distribution, the sample variance (scaled appropriately) follows a chi-squared distribution. This relationship is often utilized to establish probabilities related to the sample standard deviation or variance, using degrees of freedom based on the sample size.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It quantifies how much the sample mean is expected to vary from one sample to another. The standard error is calculated by dividing the population standard deviation by the square root of the sample size, thereby linking the variability in individual observations to the precision of the sample mean as an estimator of the population mean.

Z-score

A Z-score is a statistical measure that describes a value's position relative to the mean of a group of values, measured in terms of standard deviations. In the context of sample means, Z-scores are used to standardize values, which allows us to use the standard normal distribution to determine probabilities and percentiles.

Sampling Distribution of the Sample Mean

When samples of a fixed size are drawn from a population, the distribution of the sample means tends to be normally distributed—especially if the original population is normal. This concept is crucial because it allows us to make inferences about the population mean using the Central Limit Theorem, even when dealing with sample data.

Normal Distribution

This is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by its mean (which determines the center of the distribution) and its standard deviation (which determines the spread). Many real-world variables, including test scores under certain conditions, are modeled using the normal distribution, allowing for probability calculations and predictions about population parameters.

Central Limit Theorem

This fundamental statistical theorem states that, regardless of the shape of the original population distribution, the distribution of the sample means will approach a normal distribution as the sample size increases. This theorem justifies the use of normal probability methods for sample means, particularly when the sample size is sufficiently large, even if the original data are not normally distributed.

Transcript

00:01 Okay, so here we can say that using the central limit theorem, we can say that z is equal to x bar minus 420, all over 20 is going to be asymptotically normally distributed with mean zero and standard deviation 1.

00:15 So then in part a, we want to compute the probability if the sample mean score is higher than 450, so we want the probability that x bar is greater than 450, which this is going to be equal to 1 minus the probability that z is less than equal to 1 .5.

00:40 So this can be equal to 1 minus 0 .9332, which is going to give us the probability here is equal to 0 .0668.

00:55 So that's going to give us the probability that the sample mean score is higher than, 450, that's going to be equal to 0 .0668.

01:10 And then in, and then we want to, in part b, we're interested in computing the probability that we have 400, less than or equal to x bar, which is less than or equal to 450.

01:29 So this is going to be equal to the probability that z is less than equal to 1 .5 minus the probability that z is less than equal to negative 1.

01:43 So that gives us 0 .9332 minus 0 .1587, giving us the probability here that the sample mean score is between 400 and 450 is going to be equal to 0 .7745.

02:01 And then in part c, we want to find the value above which 10 % of the sample means of the scores lie.

02:11 So here we want the probability that x bar minus, was less than or equal to lowercase x bar.

02:22 Here it could be equal to 0 .9.

02:25 So we have the, from our normal table, we know the probability that z is less than equal to 1 .28 is going to be equal to 0 .9.

02:37 So here we have x bar minus 420 over 20 is equal to 1 .28, giving us that x bar is then equal to 1 .28 times 20 and then plus 420.

02:49 So that's going to be equal here to 445 .63.

02:58 And for part d, we want to find the value below, which 10 % of the sample means of the scores lie.

03:08 So here we get the probability that z is less than equal to x bar minus 420, all over 20 is then equal to 0 .1.

03:19 From our number of table, we get the probability that z is less than equal to negative 1 .28 is 0 .1.

03:26 So then we have that x bar minus 420, all divided by 20, is going to be equal to negative 1 .28, giving us that x bar is equal to negative 1 .28 times 20 plus 420, which gives us 394 .37.

03:46 So that's going to be the value below which 10 % of the sample means, of the score lie is again 394 .37...

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