# Math 14 HW 6.4.9-T Using the Central Limit Theorem (2023)

## Introduction

Find the probability that 1 randomly selected adult male has a weight greater than 154 lb. Find the probability that a sample of 28 randomly selected adult males has a mean weight greater than 154 lb. What do you conclude about the safety of this​ elevator?

## Content

An elevator has a placard stating that the maximum capacity is 4 300 pounds, which is 28 passengers.

So 28 adult male passengers can have a mean weight of up to 4 300 divided by 28, which is equal to 154 pounds.

Assume that weights of males are normally distributed with a mean of 183 pounds and a standard deviation of 29 pounds for part A.

It says, find the probability that one randomly selected adult male has a weight greater than 154 pounds.

Okay.

So notice that an individual value from a normally distributed population has been chosen.

So therefore we're going to use the population distribution to determine the probability.

So let's, first, draw our picture.

Okay.

So we're going to sketch our bell curve and then we're going to make note of the mean.

And the mean says that the assume the weights the male and organ distribute with a mean of 183 pounds.

So this mean is 183, and we know that the standard deviation is given to be 29 pounds.

Okay.

Now in the question for part, A, it says, find the probability that one randomly selected adult has a weight, that's greater than 154.

Well, we first want to label where 154 is on our diagram, which is to the left of 183.

So we would say that X is equal to 154.

And what we want to do is we want to find the probability of when the weight is greater than that.

So we're looking for the area that's being shaded to the right here.

Okay.

So in order to do that what we need to do is find our z-score so we're going to convert 154 to a z-score.

So we know the formula for the z-score is equal to X.

Well, first let's write down the given information.

So we know that X is equal to 154.

We know that our mean is 183, and we know the standard deviation is 29.

So now we can find that z-score, which is the formula x minus the mean over the standard deviation.

So X is 154, we're subtract an a mean of 183 and then we're going to divide that by 29.

So let's, go ahead and do that on our calculator.

So we take 154 subtract 183 and then we're going to divide that by 29, and that gives us negative 1.00.

So that tells us what the z-score is for that value.

So underneath here, we're going to draw our z-axis where the mean is zero now.

And now this z-score for the x value of 154 is negative 1.00.

And so what we want to do is we want to find the probability of when that z-score is going to be greater than or equal to negative 1.00.

Okay.

So in order to do that we're going to go ahead and use statcrunch so let's open up statcrunch.

And then we want to do is we want to open up the normal calculator.

So we'll go to calculators and scroll all the way down to the normal distribution calculator.

Okay.

And so what we're concerned is is we want to be able to find the z-scores is greater than or equal to negative 1.00.

We know, the mean is zero.

The standard deviation is one we're going to change the inequality to point to the right and then we're going to put in negative 1.00 and then select compute.

And now it's given our area and we're going to round that to four decimal places.

So we get 0.8413 0.8413.

And then that represents the area that's shaded to the right so let's, go ahead and put that in there to check so 0.8413.

And there is our result.

Now it says, find the probability that a sample of 28 randomly selected.

Adult males has a mean weight greater than 154 pounds.

Okay.

So in this case, the desired probability is for the mean of a sample of 28 adult male passengers, therefore we're going to use the central limit theorem.

Now, the central limit theorem applies when a population has a normal distribution or the sample size n is greater than 30.

Well, in this case here it says that it's normally distributed.

So we don't have to be worried about the fact that it's greater than 30.

So let's, go ahead and draw our picture.

First, okay.

So I'm going to draw a picture.

And again, we're going to have the same looking distribution here where the mean is given to be 183, okay.

And we know that the value has to still be greater than 154.

So that means to the left of the mean we have an x value, which equals 154.

And then it says, they want us to find the probability of it being greater than.

So we want to find the value that is greater so we're going to shade to the right.

Okay.

Now, keep in mind here that since we're using the central limit theorem.

This mean, is the mean of the sample means so we're going to write down the given information so that we can then apply it.

So what do we know? We know, the value of x is equal to 154? Okay, we know that the mean of the sample means is equal to the mean, which is 183, we know the sample size for Part B is 28 so that's 28 randomly selected adult males.

And then we want to find the standard deviation with the sample means.

Now, remember that this is the standard deviation divided by the square root of n, and that standard deviation is 29 and then we're going to divide that by the square root of 28.

now, we're going to plug that into our formula.

So let's, go ahead and do that.

Okay, so we know that the z-score is going to equal the value of x minus the mean of the sample means divided by the standard deviation of the sample means so we're going to plug that in.

So we got 154 minus 183 and then we're going to divide that by 29 over the square root of 28.

And then we want to determine what is that z-score.

So again, using parentheses is going to help when you're doing your calculation.

So in the numerator, we have parentheses 154 minus 183 and then we're going to divide that by parentheses 29 divided by the square root of 28 and then we're going to close the parentheses and then press enter.

And now we get a z-score of negative negative 5.29.

So we get negative 5, .29.

Okay.

So what does that tell us? Well now we can rewrite the z-axis.

So we know that the mean is going to be zero, and we know that our z-score with the central limit theorem is going to be negative 5.29.

So now what we want to do is we want to find the probability of when the z-score is going to be greater than or equal to negative 5.29.

So now let's go ahead and then use statcrunch to determine that result.

So we have our mean of zero standard deviation of one.

We want to make sure it's greater than or equal to and then we're going to put in our z-score of negative 5.29 and then select compute.

And you can see here that we end up getting 0.9999999 foreign places.

So if we round up to four decimal places, that means we got to change all these nines to zeros.

And then this last zero on the left is one so it's going to be one point zero zero, zero, zero.

So let's, go ahead and put that probability in here.

We get one point zero zero.

Zero, zero, press enter and then there's our result.

Now, the next question says, does the elevator appear to be safe? Well, in this case, we're going to say, no, because there is a good chance that 28 randomly selected adult male passengers will exceed the elevator capacity.

So we're going to go ahead and select B.

And there is our result.

## FAQs

### What is 6.4 the central limit theorem? ›

Roughly, the central limit theorem states that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution.

How do you answer the central limit theorem? ›

The Central Limit Theorem and Means

In other words, add up the means from all of your samples, find the average and that average will be your actual population mean. Similarly, if you find the average of all of the standard deviations in your sample, you'll find the actual standard deviation for your population.

What is the central limit theorem for the t test? ›

Central Limit Theorem in the Student's t-test

Since the central limit theorem determines the sampling distribution of the means with a sufficient size, a specific mean ( X ̅ ) can be standardized z = X - - µ σ n and subsequently identified against the normal distribution with mean of 0 and variance of 12.

What is the central limit theorem for high school students? ›

The Central Limit Theorem

Formally, the CLT says: If samples of size are drawn at random from any population with a finite mean and standard deviation, then the sampling distribution of the sample means, , approximates a normal distribution as increases.

Is central limit theorem easy? ›

The central limit theorem in statistics states that, given a sufficiently large sample size, the sampling distribution of the mean for a variable will approximate a normal distribution regardless of that variable's distribution in the population. Unpacking the meaning from that complex definition can be difficult.

What is central limit theorem basics? ›

The central limit theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement , then the distribution of the sample means will be approximately normally distributed.

What is the central limit theorem example? ›

Example: Central limit theorem A population follows a Poisson distribution (left image). If we take 10,000 samples from the population, each with a sample size of 50, the sample means follow a normal distribution, as predicted by the central limit theorem (right image).

What is the central limit theorem simple examples? ›

Biologists use the central limit theorem whenever they use data from a sample of organisms to draw conclusions about the overall population of organisms. For example, a biologist may measure the height of 30 randomly selected plants and then use the sample mean height to estimate the population mean height.

What is the central limit theorem quizlet? ›

The central limit theorem states that the sampling distribution of any statistic will be normal or nearly normal, if the sample size is large enough.

What type of test is at test? ›

A t test is a statistical test that is used to compare the means of two groups. It is often used in hypothesis testing to determine whether a process or treatment actually has an effect on the population of interest, or whether two groups are different from one another.

### What is central limit theorem valid for? ›

CLT is only valid for distribution whose variance is finite. It means for distribution with infinite variance, such as the Cauchy distribution, the result of CLT will not hold. All in all, I hope this article helps you to understand the central limit theorem!

How do you calculate sample mean? ›

The general sample mean formula for calculating the sample mean is expressed as x̄ = ( Σ xi ) ÷ n. Here, x̄ denotes the average value of the samples or sample mean, xi refers all X sample values and 'n' stands for the number of sample terms in the given data.

What does CLT mean in algebra? ›

Key Takeaways. The central limit theorem (CLT) states that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the population's distribution. Sample sizes equal to or greater than 30 are often considered sufficient for the CLT to hold.

Why is central limit theorem important? ›

The Central Limit Theorem is important for statistics because it allows us to safely assume that the sampling distribution of the mean will be normal in most cases. This means that we can take advantage of statistical techniques that assume a normal distribution, as we will see in the next section.

When can you not use the central limit theorem? ›

If the sample size is at least 30 or the population is normally distributed, then the central limit theorem applies. If the sample size is less than 30 and the population is not normally distributed, then the central limit theorem does not apply.

What is a good sample size? ›

Many statisticians concur that a sample size of 100 is the minimum you need for meaningful results. If your population is smaller than that, you should aim to survey all of the members. The same source states that the maximum number of respondents should be 10% of your population, but it should not exceed 1000.

How do you calculate Z score? ›

The z-score of a value is the count of the number of standard deviations between the value and the mean of the set. You can find it by subtracting the value from the mean, and dividing the result by the standard deviation.

What is central limit theorem PDF? ›

According to the central limit theorem, the means of a random sample of size, n, from a population with mean, μ, and variance, σ², distribute normally with mean, μ, and variance, σ²/n.

What is n in statistics? ›

The symbol 'n,' represents the total number of individuals or observations in the sample.

Why is 30 a good sample size? ›

It's that you need at least 30 before you can reasonably expect an analysis based upon the normal distribution (i.e. z test) to be valid. That is it represents a threshold above which the sample size is no longer considered "small".

### What are the rules of limit theorem? ›

Power law for limits states that the limit of the nth power of a function equals the nth power of the limit of the function. Root law for limits states that the limit of the nth root of a function equals the nth root of the limit of the function.

What is meant by 95% confidence interval? ›

Strictly speaking a 95% confidence interval means that if we were to take 100 different samples and compute a 95% confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the true mean value (μ).

How to find standard error? ›

How do you calculate standard error? The standard error is calculated by dividing the standard deviation by the sample size's square root. It gives the precision of a sample mean by including the sample-to-sample variability of the sample means.

What is central limit theorem for kids? ›

In probability theory and statistics, the central limit theorems, abbreviated as CLT, are theorems about the limiting behaviors of aggregated probability distributions. They say that given a large number of independent random variables, their sum will follow a stable distribution.

What are the two principles of the central limit theorem? ›

The Central Limit Theorem's Properties

Normal distributions have two parameters: mean and standard deviations. As the sample size grows, the sample distribution's amplitude comes together on a normal distribution where the means equals the population mean, and the standard deviation equals σ/√n.

What is the formula for the standard normal distribution? ›

The standard normal distribution (z distribution) is a normal distribution with a mean of 0 and a standard deviation of 1. Any point (x) from a normal distribution can be converted to the standard normal distribution (z) with the formula z = (x-mean) / standard deviation.

Which of the following is the formula for an expected value? ›

To find the expected value, E(X), or mean μ of a discrete random variable X, simply multiply each value of the random variable by its probability and add the products. The formula is given as E ( X ) = μ = ∑ x P ( x ) .

What is a high t-value? ›

Higher values of the t-score indicate that a large difference exists between the two sample sets. The smaller the t-value, the more similarity exists between the two sample sets.

What is the T-score in at test? ›

A t-score (a.k.a. a t-value) is equivalent to the number of standard deviations away from the mean of the t-distribution. The t-score is the test statistic used in t-tests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a t-distribution.

What is the t-test for college? ›

Definition/Introduction

 In simple terms, a Student's t-test is a ratio that quantifies how significant the difference is between the 'means' of two groups while taking their variance or distribution into account.

### What is the sample size for a large population? ›

A good maximum sample size is usually around 10% of the population, as long as this does not exceed 1000. For example, in a population of 5000, 10% would be 500. In a population of 200,000, 10% would be 20,000. This exceeds 1000, so in this case the maximum would be 1000.

How does sample size matter? ›

A sample that is larger than necessary will be better representative of the population and will hence provide more accurate results. However, beyond a certain point, the increase in accuracy will be small and hence not worth the effort and expense involved in recruiting the extra patients.

What is the limit of probability? ›

In Bayesian inference, or Bayesian statistics, probability limits are also referred to as “credibility limits.” Probability limits are the upper and lower end-points of the probability (or credible) interval that has a specified (posterior) probability (e.g., 95% or 99%) of containing the true value of a population ...

Does sample mean free? ›

A sample is a small quantity of a product, given free so that customers can try it or examine it before making the decision to buy.

What is the mean example? ›

Mean: The "average" number; found by adding all data points and dividing by the number of data points. Example: The mean of 4, 1, and 7 is ( 4 + 1 + 7 ) / 3 = 12 / 3 = 4 (4+1+7)/3 = 12/3 = 4 (4+1+7)/3=12/3=4left parenthesis, 4, plus, 1, plus, 7, right parenthesis, slash, 3, equals, 12, slash, 3, equals, 4.

How to find the median? ›

The median is the middle value in a set of data. First, organize and order the data from smallest to largest. To find the midpoint value, divide the number of observations by two. If there are an odd number of observations, round that number up, and the value in that position is the median.

What does C stand for in algebra? ›

The Latin small letter c is used in math to represent a variable or coefficient.

What are ABC called in algebra? ›

The ABC formula is — you guessed it — a formula, specifically used for solving quadratic equations. The name “ABC” comes from the coefficients of the quadratic equation, written in standard form: a x 2 + b x + c = 0.

What is the central limit theorem for variables? ›

The Central Limit Theorem (CLT) says that the distribution of a sum of independent random variables from a given population converges to the normal distribution as the sample size increases, regardless of what the population distribution looks like.

What is one of the most important points of the central limit theorem? ›

One of the most important components of the theorem is that the mean of the sample will be the mean of the entire population. If you calculate the mean of multiple samples of the population, add them up, and find their average, the result will be the estimate of the population mean.

### What is the difference between a population and a sample? ›

A population is the entire group that you want to draw conclusions about. A sample is the specific group that you will collect data from.

What happens when sample size increases? ›

The larger the sample size, the more accurate the average values will be. Larger sample sizes also help researchers identify outliers in data and provide smaller margins of error.

What is 7.2 central limit theorem? ›

The central limit theorem for sample means says that if you keep drawing larger and larger samples (such as rolling one, two, five, and finally, ten dice) and calculating their means, the sample means form their own normal distribution (the sampling distribution).

What is 7.3 the central limit theorem? ›

The central limit theorem for sums says that if you keep drawing larger and larger samples and taking their sums, the sums form their own normal distribution (the sampling distribution), which approaches a normal distribution as the sample size increases.

Can central limit theorem be greater than 30? ›

The central limit theorem (CLT) states that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the population's distribution. Sample sizes equal to or greater than 30 are often considered sufficient for the CLT to hold.

What is an example of the central limit theorem? ›

Biologists use the central limit theorem whenever they use data from a sample of organisms to draw conclusions about the overall population of organisms. For example, a biologist may measure the height of 30 randomly selected plants and then use the sample mean height to estimate the population mean height.

What is the limit theorem 7? ›

The central limit theorem states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed.

What is central limit theorem 200? ›

The Central Limit Theorem states that if the sample size is sufficiently large then the sampling distribution will be approximately normally distributed for many frequently tested statistics, such as those that we have been working with in this course: one sample mean, one sample proportion, difference in two means, ...

Why 30 samples for normal distribution? ›

It's that you need at least 30 before you can reasonably expect an analysis based upon the normal distribution (i.e. z test) to be valid. That is it represents a threshold above which the sample size is no longer considered "small".

Why is it called central limit theorem? ›

2) "Central" comes from "fluctuations around centre (=average)", and any theorem about limit distribution of such fluctuations is called CLT.

### Why is the central limit theorem important quizlet? ›

The Central Limit Theorem is important in statistics, because: For a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the distribution of the population.

What is the limit theorem rule? ›

The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal.

What is the 10% condition central limit theorem? ›

The 10% Condition: When the sample is drawn without replacement, the sample size should be no larger than 10% of the population.

What if n is greater than 30 normal? ›

The general rule is that if n is more than 30, then the sampling distribution of means will be approximately normal. However, if the population is already normal, then any sample size will produce a normal sampling distribution.

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