STAT 121: Lesson 14: Sampling Distribution of X-bar & The Central Limit Theorem

Sampling Distribution of X-bar

A distribution of the sample mean; a list of all the possible values for x-bar together with the frequency (or probability) of each value.

If a large population has mean, 𝜇=40μ=40 and standard deviation, 𝜎=12,σ=12, what is the standard deviation of the sampling distribution of 𝑥¯x¯ created from all possible samples of size 𝑛=9?

4

If a large population has mean, 𝜇=40μ=40 and standard deviation, 𝜎=12,σ=12, what is the mean of the sampling distribution of 𝑥¯x¯ created from all possible samples of size 𝑛=9?

40

How do you decide whether a result is a statistic or a parameter?

Be determining whether the result is from a sample or the entire population.

Why could we use the standard Normal table to find a probability on a randomly selected bottle weighing more than 1.11.1 pounds?

Because the population distribution is Normal.

Sampling Distribution of X-bar: Center

The mean of the sampling distribution of x-bar = the population mean, u, mu.

What does the Central Limit Theorem allow us to do?

Compute a probability on x-bar using a Normal curve if the sample is large & random.

In context, what parameter do we want to estimate?

The mean yield of the new corn variety in bushels per acre.

True or false: If a random sample is taken, the shape of the sampling distribution of 𝑥¯x¯ is closer to Normal for smaller sample sizes than for larger sample sizes.

False

True or false: The larger the sample size, the farther 𝑥-bar will be from 𝜇.

False

Why is the sampling distribution so important?

If a sampling distribution has a lot of variability, then if you took another sample it's likely you would get a very different result. About 95% of the time. the sample mean will be within 2(sigma/sq root of n) of the population mean. This tells us how "close" the sample statistic should be to the population parameter.

Center

Mean of sampling distribution of x-bar is u mu.

Closing prices of stocks have a right-skewed distribution with u = 26 & o = 20. What is the probability of randomly selecting a closing stock whose value is less than $15? What is the distribution of closing stock prices? (Individual, not sample).

Right-skewed. n < 30. Don't use the CLT.

Why do we care about the sampling distribution?

Sampling distributions allow us to assess uncertainty of sample results. If we know the spread of the sampling distribution, we would know how far our x-bar might be from the true mu (one way or another).

Spread

Standard deviation of sampling distribution of x-bar is o theata / sq root of n.

What is the basic idea of the law of large numbers?

The larger the sample size, the closer the sample mean is to the population mean.

True or false: We could compute both a probability on the weight of a randomly selected individual bottle and a probability on the mean weight of a random sample of eight bottles using the standard Normal table because weights of the bottles is Normally distributed.

True

The sampling distribution of x-bar gives _______ from all possible samples of the same size from the same population.

a) All x-bar values.

Consider taking a random sample of size 49 from a left-skewed population with mean 80 and standard deviation 7. What is the standard deviation of the sampling distribution of sample means for n = 49 from this population?

b) 1

Consider taking a random sample of size 49 from a left-skewed population with mean 80 and standard deviation 7. What is the mean of the sampling distribution of sample means for n = 49 from this population?

b) 80

True or false: In inference we use the value of a parameter to estimate a statistic.

false

Parameter

numerical fact about a population. Example: mu = average GPA of full-time UVU students.

When can we compute a probability on a sample mean, x-bar, using Normal distribution?

when the sampling distribution of x-bar is either Normal or approximately normal.

If a large population has a Normal shape with mean, 𝜇=40μ=40 and standard deviation, 𝜎=12,σ=12, what is the shape of the sampling distribution of x-bar created from all possible samples of size n = 9?

Normal

Sigma O

Population standard deviation.

The BYU Creamery sells bottles of chocolate milk with a mean weight of 1.0875 pounds (u, mu) & a standard deviation (o, sigma) of 0.015 pounds. The weights of these bottles are Normally distributed: What is the probability that a randomly selected bottle weights more than 1.1 pounds?

Steps: 1) Draw & label the Normal curve. 2) Compute appropriate Z-score: Given X to Z to Area. Z = x-u, mu / sigma, o. So then, Z = 1.1-1.0875/0.015 = 0.83. 3) Look-up z-score on table to get P(x>1.1) = P(z>0.83). Then, subtract from one: 1-0.7967 = 0.2033. The probability that a randomly selected bottle weighs more than 1.1 pounds is 0.2033.

The Sampling Distribution of X-bar

The sampling distribution of a sample mean (X-bar) is a theoretical probability distribution. It describes the distribution of: all sample means, from all possible random samples, of the same size, taken from the same population.

Why is the normal distribution so important? Central Limit Theorem

The sampling distribution of a statistic (like a sample mean) often follows a normal distribution if the sample sizes are large.

According to the Central Limit Theorem, what has an approximate Normal shape if the sample is large and random?

The sampling distribution of x-bar.

Sampling Distribution of X-bar: Case 2: Population Non-normal

The shape of the sampling distribution of x-bar is approximately Normal when n is large.

Central Limit Theorem

The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution. CLT allows us to use the standard normal table to compute approximate probabilities associated with x-bar.

True or false: The mean of the sampling distribution of x-bar equals u, mu regardless of sample size.

True

True or false: The spread of the sampling distribution of x-bar decreases as n increases.

True

True or false: The value of 𝑥-bar varies, changes, from one sample to the next.

True

If the response variable is quantitative, do we estimate a mean or a proportion?

a mean

If the shape of the population distribution is Normal, what is the shape of the sampling distribution of x-bar?

always Normal regardless of sample size

Why is the sampling distribution of x-bar approximately Normal?

because the sample is large and random, allowing us to apply the Central Limit Theorem

Means of random samples are

less variable than individual observations.

Mean X-bar

mean of the sampling distribution of x-bar.

What symbol represents the standard deviation of a sample?

s

s

sample standard deviation

SD X-bar

standard deviation of the sampling distribution of x-bar.

Is the mean number of pepperoni slices on a 12" pizza from a sample of a certain brand of pepperoni pizzas a statistic or a parameter?

statistic

Is the mean of the measurements of individuals in a sample a statistic or a parameter?

statistic

Is the proportion of people who prefer Coke over Pepsi in a sample of mall shoppers a statistic or a parameter?

statistic

Sampling Variability

the natural tendency of randomly drawn samples to differ, one from another

Which z‑score formula should we use to find a probability on the mean weight of a random sample of eight bottles?

z = x-bar - u, mu/o, sigma/sq root of n

What symbol represents the mean of a population?

μ (mu)

Closing prices of stocks have a right-skewed distribution with u, mu = 26 & o, sigma = 20. What is the probability that the mean price of a random sample of n = 32 closing stocks is less than $15? What is the sampling distribution of x-bar? (sample, not individual).

Approximately normal due to n = 32. Can use the CLT. u, mean = 26 & o, sigma, standard deviation = o, sigma/sq root of n = 20/st root of 32 = 3.54. Given X-bar to Z to Area. Use: Z = x-bar - u, mu/o, sigma/sq root of n. Steps: Draw & Label the Normal curve. 2) Compute approximate z-score: z = x-bar -u, mu/o, sigma/sq root of n = 15-26/ 20/sq root of 32 = -3.11 = z-score. 3) Look-up z-score in table to get probability. 0.0009. The probability that the mean closing price of a random sample of 32 stocks is less than $15 is approximately 0.0009 or about 0.1%.

Why could we use the standard Normal table to find a probability on the mean of a random sample of eight bottles weighing more than 1.11.1 pounds?

Because the population distribution is Normal.

Why don't we need to create the sampling distribution of x-bar before computing a probability of x-bar?

Because we can use facts about the sampling distribution of x-bar to predict shape, center, & spread of the sampling distribution of x-bar.

After looking up the 𝑧z‑score in the standard Normal table, why did we subtract the table probability from 1.0?

Because we wanted the probability that a randomly selected bottle weighs more than 1.11.1 pounds.

True or false: The mean of the sampling distribution of x-bar equal u, mu only when n > 30?

False

True or false: The standard deviation of the sampling distribution of x-bar equals o, sigma regardless of sample size.

False

True or false: We should get the same value for 𝑥¯x¯ from most of our samples.

False

True or false: When sampling from a non‑Normal population, the shape of the histogram of the data in the sample gets closer to Normal as sample size increases.

False

True or false: 𝑥-bar almost always equals 𝜇.

False

Weights of bottles of chocolate milk are Normally distributed with u, mu = 1.0875 & o, sigma = 0.015. What is the probability that the mean of a random sample of eight bottles of chocolate milk, x-bar, exceeds 1.1 pounds?

Given x-bar to Z to Area. When finding a probability on 𝑥¯,x¯, we must use the standard deviation of the sampling distribution of x-bar, x-bar, namely 𝜎/sq root of n, in the denominator. Thus, the appropriate z‑score is 𝑧= x-bar −𝜇 /𝜎/sq root of n. Steps: 1) Draw & Label Normal Curve, 2) Compute appropriate z-score in table to get probability. P(x-bar>1.1) = P(z>2.36). Table. 0.9909. Then subtract from one. 1-0.9909 = 0.0091. The probability that the average of a random sample of eight bottles is more than 1.1 pounds is 0.0091.

Shape of Sampling Distribution of X-bar

If the population of scores is normally distributed, the sampling distribution of X bar will also be normally distributed, regardless of sample size

Sampling Distribution of X-bar: Summary

Means are equal for all n. Standard deviation of x-bar decreases as n increases. Shape of sampling distribution of x-bar becomes more Normal as n increases.

Can we use a Normal distribution to find a probability on the closing price of an individual stock?

No, because the population distribution is right skewed, not Normal.

Sampling Distribution of X-bar: Shape: Case I: Population Normal

The shape of the sampling distribution of x-bar is normal.

Sampling Distribution of X-bar: Spread

The standard deviation of the sampling distribution of x-bar = sigma o/ sq root of n.

True or false: The standard deviation of the sampling distribution of x-bar equals o, sigma/sq root of n regardless of sample size & population shape.

True.

If the shape of the population distribution is non‑Normal, what is the shape of the sampling distribution of x-bar?

approximately Normal for 𝑛 large and slightly skewed for 𝑛 small if the population is skewed.

If the shape of the population distribution is Normal, what is the shape of the histogram of data in a sample?

approximately Normal for 𝑛n large and unpredictable for 𝑛n small.

If the shape of the population distribution is non‑Normal, what is the shape of the histogram of data in a sample?

approximately like the population shape for 𝑛 large and unpredictable for 𝑛 small.

What is the definition of the sampling distribution of x-bar?

b) The values of x-bar from all possible samples of the same size from the same population.

When can we compute a probability on an individual 𝑥x using a Normal distribution?

c) When the population distribution is normal.

When can we compute a probability on a sample mean, x-bar, using a normal distribution?

c) When the sampling distribution of x-bar is either Normal or approximately Normal.

If a large population has mean, 𝜇=40μ=40 and standard deviation, 𝜎=12,σ=12, what is the shape of the sampling distribution of x-bar created from all possible samples of size n = 9?

cannot be determined since 𝑛=9n=9 is too small for case 22 and we do not know whether the population shape is Normal

Consider taking a random sample of size 49 from a left-skewed population with mean 80 and standard deviation 7. Suppose we're planning to take a random sample of size 49 from this population. What sample mean value are we expecting to get?

d) An x-bar close to 80

Population Distribution

how population is spread out in an area

Means of random samples are

more normal than individual observations.

Statistic

numerical fact about the sample. Example: x-bar = average GPA of the 200 UVU students in our sample.

What symbol represents the standard deviation of a population?

o, sigma

What symbol represents the standard deviation of the sampling distribution of x-bar?

o, sigma/sq root of n

Is the mean SAT of all entering freshmen a statistic or a parameter?

parameter

Is the proportion of individuals in a population with a particular characteristic a statistic or a parameter?

parameter

Sample

part of a population. A subgroup of the population from which we obtain information. Example: 200 full-time UVU students.

u (mu)

population mean

Yield is to be measured as the response variable. Is it categorical or quantitative?

quantitative

X-bar

sample mean

Population

the entire group of individuals that is the target of our interest. Example: All full-time UVU students.

What symbol represents the mean of the sampling distribution of x-bar?

u, mu

What symbol represents the mean of a sample?

x-bar

Which z‑score formula should we use to find a probability on an individual bottle?

z = x - u, mu/o,sigma

When sampling from a non-Normal population with a large random sample, what 𝑧z‑score formula should we use to find a probability on ?

z = x-bar - u, mu/o, sigma/sq root of n