The population of IQ scores forms a normal distribution with a mean of μ=100 and a standard devi

The population of IQ scores forms a normal distribution with...

The population of $\mathrm{IQ}$ scores forms a normal distribution with a mean of $\mu=100$ and a standard deviation of
$\sigma=15$. What is the probability of obtaining a sample mean greater than $M=97$,
a. for a random sample of $n=9$ people?
b. for a random sample of $n=25$ people?

Key Concepts

Normal Distribution

A normal distribution is a symmetric, bell-shaped probability distribution characterized by its mean and standard deviation. It is a fundamental distribution in statistics because many variables naturally follow this pattern, and numerous statistical methods assume that data are normally distributed.

Central Limit Theorem

The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normally distributed, regardless of the population’s distribution, provided that the sample size is sufficiently large. This theorem justifies using the normal model for the sample mean even when sampling from a non-normal population.

Standard Error

The standard error is the measure of the variability of the sample mean from sample to sample. It is calculated by dividing the population standard deviation by the square root of the sample size. A larger sample size decreases the standard error, indicating that the sample mean is likely to be closer to the true population mean.

Z-Score Transformation

Z-score transformation is a method of standardizing a value by subtracting the mean and dividing by the standard deviation. When applied to the sample mean, it allows the comparison of the observed value with the standard normal distribution to calculate probabilities associated with that observation.

Transcript

00:01 In this question, the motive is to calculate the value of probability such that the random variable x follows normal distribution and the values of parameter such as mu is equals to 100 and the value of sigma that is standard deviation is equals to 15.

00:19 Now we can say that in the first part the value of probability such that x is greater greater than 140 will be here equals to probability of x minus mu divided by sigma and this is here greater than 140 minus it is 100 divided by it will be 15 and hence we can say that the value of probability will be here z and that is greater than 2 .67 which can be here written to be equals to 1 minus probability of z and that is lesser than 2 .67.

00:59 So this very value will be here equals to 1 minus 0 .9962 with the help of excel command that we will write in just a minute.

01:11 Now it is here 0 .0038.

01:14 So this is the required answer in the first part.

01:17 Now let us let's discuss the excel command that we can use.

01:21 So the excel command says that probability of z less than a where a can be negative or positive can be calculated with the help of the command which is equal norm dot s dot it will be dist and here the value of a will be present and now it will be true.

01:44 True.

01:44 So using this very excel command we can calculate the probability of z less than a and we will use this concept in our problem which are going to be solved further.

01:56 Let us see how we can do so and in the second part the value of probability such that it is 120 and here it is x now it will be 140 so this is equals to probability of let us write 120 minus 100 divided by the sigma value which is given in the problem as 15.

02:19 Now it is here z less than 140 minus 100 divided by 15 and if we simplify this very expression then it will be here probability of 1 .33 and that is lesser than z and it will be lesser than 2 .67.

02:39 So this this is here equals to probability of z and that is lesser than 2 .67 minus it will be probability of z and that is lesser than 1 .33...

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