An instructor gives a 100-point examination in which the...

Name: Problem 38, Chapter 6: Elementary Statistics a Step by Step Approach
Uploaded: 2020-06-04T05:16:08-08:00
Duration: 5 min 52 s
Channel: Emily Miller
Description: An instructor gives a 100-point examination in which the grades are normally distributed. The mean is 60 and the standard deviation is 10. If there are 5% A’s and 5% F’s, 15% B’s and 15% D’s, and 60% C’s, find the scores that divide the distribution into those categories.

An instructor gives a 100-point examination in which the grades are normally distributed. The mean is 60 and the standard deviation is 10. If there are 5% A’s and 5% F’s, 15% B’s and 15% D’s, and 60% C’s, find the scores that divide the distribution into those categories.

Key Concepts

Percentiles

Percentiles represent the value below which a given percentage of observations in a group of observations falls. They are used to determine cut-off scores that segment the distribution into different categories or intervals, which is particularly useful in grading, performance evaluation, and risk management.

Standardization (Z-score Transformation)

Standardization involves converting a normally distributed variable with any mean and standard deviation to a standard normal variable (Z-score) which has a mean of 0 and a standard deviation of 1. This process simplifies the calculation of probabilities and percentiles using the standard normal distribution tables or functions.

Inverse Cumulative Distribution Function (Quantile Function)

The inverse cumulative distribution function, or quantile function, is used to determine the value of a variable that corresponds to a specific cumulative probability. In the context of a normal distribution, it allows one to calculate the threshold scores or cut-off values that correlate with given percentiles.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence. It’s characterized by its bell-shaped curve and is commonly used in statistics due to properties such as the empirical rule and the central limit theorem, which make it a powerful tool for modeling and inference.

Mean and Standard Deviation

The mean provides the central or average value of a data set, while the standard deviation measures the dispersion or spread of the data around the mean. In a normal distribution, these two parameters fully determine the shape and scale of the distribution.

Transcript

00:01 In this problem, we're analyzing a set of test scores out of 100 points that are normally distributed, have a mean of 60, and a standard deviation of 10.

00:11 We're given the percent breakdowns of the scores, which i've shown here on a standard bill curve, and we're asked to find the scores that divide the distributions into these categories, meaning at what range of scores would a student earn an a, b, c, d, or f? let's start with our bottom 5 percent.

00:35 X sub f or the score at which someone would earn an f.

00:40 So the z score below which 5 % of our test scores lie is a negative 1 .64, which i've rounded to the hundredth place.

00:49 Taking into consideration what i know, which is a z score, a standard deviation, and a mean, the z score formula is automatically brought to mind.

00:59 So if you'll recall, z score equals a raw score, which is what we're looking for in this case.

01:06 Minus the mean divided by the standard deviation.

01:11 Since we're looking for x or the raw score, we can rearrange this formula to find one that's more useful to us, one that solves for x.

01:23 So x or raw score equals the z score times the standard deviation plus the mean.

01:33 This is what we're going to use for each of these categories, starting with x sub f so x sub f equals negative 1 .64 times 10 plus 60 when we put that into our calculator we get 43 .6 so if you score below a 43 .6 on the test you're getting an f.

02:05 Moving up to x sub d so the z score below which 20 % of our test.

02:12 Scores lie is negative 0 .84 again rounded to the hundredth place.

02:18 You might be asking why do you have to take into consideration 20 % when x sub d or d's on the test only account for 15%.

02:27 Well you have to take into consideration that not only do our d scores make up this portion of our bill curve but we have to take in that bottom 5 % as well when we're considering z scores...

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