ACT scores The composite scores of individual students on...

ACT scores The composite scores of individual students on the ACT college entrance examination in 2009 followed a Normal distribution with mean 21.1 and standard deviation 5.1 (a) What is the probability that a single student randomly chosen from all those taking the test scores 23 or higher? Show your work. (b) Now take an SRS of 50 students who took the test. What is the probability that the mean score $\overline{x}$ of these students is 23 or higher? Show your work.

ACT scores The composite scores of individual students on the ACT college entrance examination in
2009 followed a Normal distribution with mean 21.1 and standard deviation 5.1
(a) What is the probability that a single student randomly chosen from all those taking the test scores
23 or higher? Show your work.
(b) Now take an SRS of 50 students who took the test. What is the probability that the mean score $\overline{x}$ of these students is 23 or higher? Show your work.

Key Concepts

Sampling Distribution of the Mean

The sampling distribution of the mean refers to the probability distribution of sample means obtained from repeated random samples of the same size from a population. This distribution has the same mean as the population but a reduced standard deviation, known as the standard error, which quantifies the variability of the sample means.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve. It is defined by its mean and standard deviation, and many natural phenomena, such as test scores, follow this distribution. It is used to calculate probabilities associated with values, especially when the distribution is symmetric and follows predictable behavior.

Standard Score (z-score)

The standard score, or z-score, is a numerical measurement that describes a value's position relative to the mean of a group of values, measured in terms of standard deviations from the mean. It allows for the standardization of different data points which can then be used to compute probabilities using the standard normal distribution tables or relevant software.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the original distribution of the population, the distribution of the sample mean will tend to be approximately normally distributed if the sample size is sufficiently large. This concept underpins many inferential statistics procedures and justifies using the normal distribution to compute probabilities for sample means.

Transcript

00:01 This video deals with the composite scores on the act test.

00:07 And our population data is given the following information.

00:12 We have a normal distribution.

00:14 We know that the mean of the normal distribution is equal to 21 .1.

00:21 And we know there's a 5 .1 standard deviation.

00:25 So on either side, i'm going to put these to one standard deviations.

00:30 And this distance right here, standard deviation is equal to 5 .1.

00:39 Now, if we figure out what numbers that this is, if i take 21 .1 and add 5 .1, i get 26 .2.

00:50 And if i take 21 .1 and subtract 5 .1, i get 16.

00:58 So part a wants me to find the probability that a randomly chosen student scores 23 or higher.

01:08 So the probability that some randomly chosen student scores 23 or higher.

01:17 So i have to estimate 23.

01:20 It's going to be somewhere right in here.

01:25 It's going to be less than one standard deviation away.

01:28 So i need to figure out what that standard deviation is.

01:31 So to find the standard deviation, i'm going to use the z score, which is the number you're looking for, so that's 23.

01:37 Minus the mean, so that's 21 .1 divided by the standard deviation, which is 5 .1.

01:46 That's going to equal to 0 .3725.

01:53 So that's my standard deviation of 23.

01:58 So now i'm looking for the probability that the student score is greater than equal to 23.

02:04 That would be the same thing as the probability of a z score being greater than or equal to 0 .3725.

02:17 Now, when i look up the z score in the chart, i'm going to find, when i look up this number, 0 .3725, that gives me this area to the left.

02:35 So that's the area that i'm going to be getting.

02:40 But i am looking for this area to the right.

02:44 You remember the total area under the curve is one.

02:48 So for me to find the area in green, i have to find one minus the probability that z is going to be less than or equal to 0 .3725.

03:01 So if i look on my deep table at 0 .37, so here's 0 .3, and i go over to 7, i get 0 .64431.

03:20 So i get 1 minus 0 .64431.

03:30 And that's going to equal to 0 .331.

03:38 So the probability that x is going to be greater than or equal to 23 is equal to that.

03:49 So there's a 33 % chance that a random student drawn will have an act composite score of 23.

03:58 So part b says that we have a simple random sample of 50 students.

04:08 So that means that my sample size is equal to 50...

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