Assume that the test scores from a college admissions test...

Assume that the test scores from a college admissions test are normally distributed, with a mean of 450 and a standard deviation of 100 a. What percentage of the people taking the test score between 400 and $500 ?$ b. Suppose someone receives a score of $630 .$ What percentage of the people taking the test score better? What percentage score worse? c. If a particular university will not admit anyone scoring below $480,$ what percentage of the persons taking the test would be acceptable to the university?

Assume that the test scores from a college admissions test are normally distributed, with a mean of 450 and a standard deviation of 100
a. What percentage of the people taking the test score between 400 and $500 ?$
b. Suppose someone receives a score of $630 .$ What percentage of the people taking the test score better? What percentage score worse?
c. If a particular university will not admit anyone scoring below $480,$ what percentage of the persons taking the test would be acceptable to the university?

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by two parameters: the mean (which determines the center) and the standard deviation (which determines the spread). This concept is fundamental in statistics because many natural phenomena approximate a normal distribution.

Mean

The mean is the arithmetic average of a set of values and represents the central point of a distribution. In the context of a normal distribution, the mean is the point around which the data are symmetrically distributed. It serves as a critical parameter when describing the location of the data distribution.

Standard Deviation

The standard deviation is a measure of the dispersion or spread of a set of values around the mean. In a normal distribution, it quantifies the degree of variation from the mean, determining the width of the bell curve and affecting the probabilities of observations within a given range.

z-Score

The z-score is a standardized score that indicates how many standard deviations an individual data point is from the mean. This transformation allows for the comparison of scores from different normal distributions and is essential for calculating probabilities by referencing the standard normal distribution.

Cumulative Distribution Function (CDF) and Percentile Calculation

The cumulative distribution function (CDF) gives the probability that a random variable from a normal distribution is less than or equal to a specific value. Percentile calculations use the CDF to determine the relative standing of a score within the distribution, indicating what percentage of data falls below that score.

Tail Probability

Tail probability refers to the probability that a random variable falls in the extreme ends (tails) of the distribution, either far above the mean or far below it. This concept is often computed by taking the complement of the CDF and is useful for assessing the likelihood of extreme outcomes.

Transcript

00:01 So the test scores x are normally distributed with mean equal to 450 and standard deviation equal to 100.

00:12 Then x minus the mean divided by the standard deviation will be a standard normal with mean 0 and standard deviation 1.

00:30 And for question a, we want to know what percentage of people take in a test score between 400 and 500.

00:40 Well, that percentage is equal to the probability of a score being between 400 and 500.

00:50 And that is equal to the probability of 400 minus the mean divided by the standard deviation being less than your standard normal, less than 500 minus the mean divided by the standard deviation, and that is equal to the probability of negative 0 .50 being less than the standard normal being less than 0 .50 and that you can look up in a standard normal table.

01:27 This is going to be 1 minus 2 times the probability that z is greater than 0 .50 because it's a symmetric distribution.

01:36 And that then is 1 minus 2 times 0 .3085 because if you want the probability of being between negative 0 .5 and 0 .5 that is this probability that is then the same as taking 1 and subtracting out both of these tail probabilities which are equal in size so that is this value here and this is the red area so this then is equal to 1 minus 2 times 0 .3085 five that is 0 .383 and then for part b suppose someone receives a score of 630 what percentage of the people taking a test score better so here we have our mean of 450.

02:58 This person scores a 630.

03:02 Well, if we convert that to the standard normal, then that means what is your z -score here? well, the z -score there is is 630 minus the mean, oops, 450, divided by the standard deviation, which was 100.

03:30 So this is 1 .80...

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