10. Mensa Membership in Mensa requires an IQ score above...

10. Mensa Membership in Mensa requires an IQ score above $131.5 .$ Nine candidates take IQ tests, and their summary results indicated that their mean IQ score is $133 .$ ( $1 \mathrm{Q}$ scores are normally distributed with a mean of 100 and a standard deviation of $15 .$ ) a. If 1 person is randomly selected from the general population, find the probability of getting someone with an IQ score of at least $133 .$ b. If 9 people are randomly selected, find the probability that their mean IQ score is at least 133 c. Although the summary results are available, the individual IQ test scores have been lost. Can it be concluded that all 9 candidates have IQ scores above 131.5 so that they are all eligible for Mensa membership?

10. Mensa Membership in Mensa requires an IQ score above $131.5 .$ Nine candidates take IQ tests, and their summary results indicated that their mean IQ score is $133 .$ ( $1 \mathrm{Q}$ scores are normally distributed with a mean of 100 and a standard deviation of $15 .$ )
a. If 1 person is randomly selected from the general population, find the probability of getting someone with an IQ score of at least $133 .$
b. If 9 people are randomly selected, find the probability that their mean IQ score is at least 133
c. Although the summary results are available, the individual IQ test scores have been lost.
Can it be concluded that all 9 candidates have IQ scores above 131.5 so that they are all eligible for Mensa membership?

Key Concepts

Inference from Summary Statistics

When only summary statistics, such as the sample mean, are available, inferring individual characteristics necessitates caution. Even though the observed mean may meet a threshold criteria, such as a qualifying IQ for membership, the variability among individual scores means that the average does not guarantee that every individual meets the criterion. This concept underscores the importance of understanding population variability and the limitations inherent in using summary measures to make conclusions about individual data points.

Sampling Distribution of the Mean

The sampling distribution of the mean is the probability distribution of sample means computed from all possible samples of a fixed size drawn from a population. For a normally distributed population, the sample mean is also normally distributed with the same mean but with a reduced standard deviation, known as the standard error, which is the original standard deviation divided by the square root of the sample size. This concept is essential when assessing probabilities related to averages of multiple observations.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve. It is defined by two parameters: the mean, which determines the center of the distribution, and the standard deviation, which defines the spread or variability. Many phenomena, including IQ scores, are modeled by the normal distribution due to the Central Limit Theorem, and probabilities for any interval can be determined using standard techniques and tables once the distribution is standardized.

Z-Score Transformation

The z-score transformation is a method to standardize a value from a normal distribution by subtracting the mean and dividing by the standard deviation. This standardization allows one to use the standard normal distribution, which has a mean of 0 and a standard deviation of 1, to determine probabilities and percentiles. It is commonly used to calculate the likelihood of obtaining values above or below a certain threshold.

Transcript

00:01 This question tells us that mensa membership requires an iq score above 131 .5.

00:08 It says nine candidates take their iq tests and their summary results indicate that their mean iq score is 133.

00:16 It tells us to assume iq scores are normally distributed with a mean of 100 and with a standard deviation of 15.

00:24 And first of all in part a, we're asked to find the probability of one randomly selected person having an iq score, of at least 133.

00:34 So what we're finding here is the probability of x, a randomly selected person from the population, being less than actually we're dealing with at least.

00:45 So at least would mean that it would be greater than or equal to 133.

00:51 So our first step here is to find a z score for 133, using this red equation i have on the screen.

00:57 So using the equation, z is equal to x, which is 133, minus the mean, which is 100, divided by standard deviation, which is 15.

01:08 And then that'll give us a z score of 2 .20.

01:12 So now let's find that in our z table here on the left side of the screen.

01:16 So we'll go down to 2 .2, which is down here, over to 0 .00, up here.

01:23 When we meet at the middle, we get 0 .9861.

01:29 Now remember, our z table gives us what's less than, in this case, 133.

01:35 So this is the probability of x being less than 133.

01:39 So to find the probability of x being greater than or equal to 133, we'll have to subtract this from 1.

01:46 So 1 minus 0 .9861 will give us 0 .0139.

01:59 So there's about a 1 .39 % chance of a randomly selected person's iq being greater than or equal to 133.

02:08 Now in part b of the question, it says if nine people are randomly selected, find the probability that their average iq score will be at least 133.

02:19 So now we're dealing with the sample and that sample size n is nine.

02:24 Now since we're dealing with the sample and since we were told that our population of iq scores are normally distributed, that means we can use the central limit theorem to find our standard error of the mean.

02:38 So to do that, we take our population standard deviation, which is 15, and we divide that by the square root of our sample size n, which is 9...

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