# Ch 7.1 Central Limit Theorem for Sample Means

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## Ch 7.1 Central Limit Theorem for Sample Means

### Sample distribution of sample mean:

When sample means $$\bar{x}$$  of same size n taken from the same population, the Sample means have the following behavior:

1)  If the population distribution of X is normal, the distribution of $$\bar{x}$$ is always normal for all sample size n.

Sampling distribution of x-bar

2)  When population distribution of X is not normal, The sampling distribution x-bar tends to be a normal distribution. The distribution become closer to normal when sample size increase.

Activity to discover the Central Limit Theorem:

https://stats.libretexts.org/Bookshelves/Ancillary_Materials/02%3A_Interactive_Statistics/15%3A_Discover_the_Central_Limit_Theorem_Activity

Central Limit Theorem for Sample Mean:

For all sample of the same size n with n > 30, the sampling distribution of $$\bar{x}$$ can be approximated by a normal distribution with mean μ and standard deviation $$\sigma _{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

Note: -This applies to all distribution of x. If X is normally distributed, n > 30 is not needed. Any n will work.

-The sample should be a Simple Random Sample.

Central Limit Theorem:    $$\mu _{\bar{x}} = \mu$$,

$$\sigma _{\bar{x}} = \frac{\sigma}{\sqrt{n}$$}

Ex1 A standardized test with scores that are normally distributed with mean μ = 150 and standard deviation σ = 18. A class of 20 students take the test. The mean score $$\bar{x}$$ of the 20 students are calculated.

a)  Is the distribution of mean score  $$\bar{x}$$ of 20 students Normally distributed?

Ans: Yes because the original score is Normal.

a)  What is the mean and standard deviation of  ?

Use Central Limit Theorem:  mean = 150,  SD = $$\frac{18}{\sqrt{20}}$$ ≈ 4.0249

b) Find the probability that a student’s score is greater than 160.

Use Online Normal Calculator, Mean = 150, SD = 4.0249

c)  Find the probability that the mean score $$\bar{x}$$ of 20 students is greater than 160.

Click above, enter 160. Recalculate.  P( $$\bar{x}$$ > 160) = 0.0065

Ex2: Coke cans are filled so that the actual amounts have a mean of 12 oz and a standard deviation of 0.11 oz. The distribution of amount of coke is unknown.

a)  Is the distribution of mean amount of coke in 36 cans normally distributed?

Yes, because n > 30, according to CLT, $$\bar{x}$$ will be normally distributed.

b) What is the mean and standard deviation of  $$\bar{x}$$?

Ans: according to CLT:  $$\mu_ \bar{x}$$ = 12, $$\sigma _{\bar{x}}$$ = 0.11/√36 ≈ 0.01833

c)   Find the percent of individual coke with amount between 11.9 to 12.1 oz.

Use online Normal Calculator:  Mean = 12, SD = 0.11

Click between, enter 11.9 and 12.1, Recalculate.  P( 11.9 < x < 12.1 ) = 0.6367

63.67% of coke have amount between 11.9 oz to 12.1 oz.

d)  Find the percent of mean amount of 36 coke with between 11.9 and 12.1 oz.

Use online Normal Calculator:  Mean = 12, SD = 0.01833,

Click between, enter 11.9 and 12.1, Recalculate.  P( 11.9 < $$\bar{x}$$ < 12.1 ) = 1

100% of mean amount of 36 coke is between 11.9 and 12.1 oz.

Ex3. Annual incomes are known to have a distribution that is skewed to the right. Assume that 20

workers’ mean incomes $$\bar{x}$$  are collected.

a)   Will the distribution of mean income $$\bar{x}$$  be normally distributed?

Ans: No, since X is not normal and n < 30, CLT does not apply,  $$\bar{x}$$ may not be normally distributed.

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