Download Lecture 5 - West Virginia University

Lecture 8
Dan Piett
STAT 211-019
West Virginia University
Last Week
 Continuous Distributions
 Normal Distributions
 Normal Probabilities
 Normal Percentiles
Overview
 8.1 Distribution of X-bar and the Central Limit Theorem
 8.2 Large Sample Confidence Intervals for Mu
 8.3 Small Sample Confidence Intervals for Mu
Section 8.1
The Sampling Distribution of the Sample Mean and the Central
Limit Theorem
Distribution of the Mean
 Suppose we generate multiple samples of size n from a
population, we will get a sample mean from each group.
 These sample means will have their own distribution.
 The sample mean is a random variable with it’s own mean and
standard deviation aka. Standard error. (See notation on board)
 This is known as the Sampling Distribution of the Sample
Mean
 Some questions to think about:
 What is the shape of the sampling distribution?
 What is the mean and standard error of the sampling
distribution?
The Central Limit Theorem
 The distribution of the sample mean is determined by the
shape of the distribution of X
 X is Normal
 The distribution of the sample mean is normal
 Mean mu
 (The same as the mean of X)
 Standard error sigma/sqrt(n)
 (The standard deviation of X divided by the sample size)
 What if X is not normal?
Central Limit Theorem Contd
 So what if X is not Normal. Assume X~?
 The shape of X will depend on the sample size
 If n<20
 We cannot be certain the distribution of the sample mean. It is not
necessarily normal or even approximately normal
 If n≥20
 The Central Limit Theorem States that the distribution of the sample
mean will approach normality
 Mean mu
 (The same as the mean of X)
 Standard error sigma/sqrt(n)
 (The standard deviation of X divided by the sample size)
Examples
 Give the sampling distribution of the sample mean for the
following distributions:
1. X is normally distributed with a mean of 50 and a standard
deviation of 20. What is the distribution of the sample
mean of n = 25
2. X is Exponentially distributed with mean of 20 and
standard deviation of 10. What is the distribution of the
sample mean of n = 100?
3. X is normally distributed with a mean of 100 and a
standard deviation of 18. What is the distribution of the
sample mean of n = 9
Probabilities
 Since the distribution of the sample mean follows a normal
distribution (under the appropriate conditions) we can
calculate probabilities much like last week.
 All methods are exactly the same except now we calculate
the z-score using our new mean and standard error.
Example: Back to SAT Scores
 From last week we said that SAT Math Scores are Normally



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distributed with a mean of 500 and a standard deviation of
100.
X~N(500,100)
What is the sampling distribution of the sample mean of a
class of 25 students?
What is the probability that A RANDOMLY SELECTED
STUDENT scores above a 600 on the SAT Math section?
What is the probability that THE MEAN SCORE OF THE 25
STUDENTS is above 600?
Section 8.2
Large Sample Confidence Intervals
Point Estimator
 Suppose we do not know the true population mean. How
can we estimate it?
 We could find a sample and use a statistic such as the sample
mean to estimate it
 This is known as a point-estimate
 But it is unlikely that our sample mean is going to exactly match
the population mean even under perfect conditions
 For this reason, it is better to state that we believe the true
mean is between two numbers a and b
a<µ<b
 We can predict a and b using a confidence interval
Confidence Intervals
 Our confidence intervals will always be of the form:
 Sample Statistic ± critical value * error term
 For the population mean, our sample statistic is x-bar
 Our critical value will be either Z or t (I will explain t later)
 Our error term will be the standard error of the sample
mean
Large Sample Confidence Intervals
 Recall if n is “large” (≥20), X-bar’s dist. is approximately
normal with mean mu and standard error sigma/sqrt(n)
 95% CI
Example
 Find the 95% confidence interval for the mean SAT Math
Score for x-bar = 502, s = 8, n = 36
 502 ± 1.96*8/6
 (499.387,504.613)
 Conclusion:
 We are 95% confident that the true mean SAT Math Score is
between 499.387 and 504.631
 Always be sure to state your conclusion
Confidence Levels
 In the previous slide, we used a confidence level of 95%.
 This corresponded to a critical value of 1.96
 We commonly use 3 different confidence levels:
 90%
 Critical Value of 1.645
 95%
 Critical Value of 1.96
 99%
 Critical Value of 2.578
Notes on the Error Term
 The error term can be effected by 3 different things
The sample size, n
1.

The standard deviation, s
2.

The smaller s, the smaller the error term
The confidence level
3.


The larger n, the smaller the error term
The higher the confidence level, the larger the critical value,
the larger the error term
We cannot choose s in practice, but we can choose the
confidence level and often n
Section 8.3
Small Sample Confidence Intervals
Small Sample Confidence Intervals
 For “large” sample sizes, we have the convenience of knowing
that the distribution approaches normality
 We can use the Z table (1.645, 1.98, 2.645)
 For “small” sample sizes (<20) we have to do a little more
work and we must know that X is normal
 Our Central Limit Theorem Rules do not apply
 Two Cases
The population standard deviation is known
2. The standard deviation is unknown
1.
Case 1: Sigma is known
 Good news!
 In this case we handle our confidence intervals in the exact
same way we would for a large sample
 Example:
 Compute a 99% confidence interval for the population mean
when x-bar= 43.2, sigma = 18, n = 16
 (31.6, 54.8)
 We are 99% confident that the population mean is between
31.6 and 54.8
Case 2: Sigma is unknown
 When the population standard deviation is unknown we need
to make a slight adjustment to our formula.
 The adjustment is in the critical value. Rather than using our
Z values (1.645, 1.98, 2.645) we will be using t values
 t values come from the t-distribution. You will only need to
know the t-distribution for inference, not probability.
T-distribution
 t values can be found on Table F
 Notes about t
 t is mound shaped and symmetric about the mean, 0
 Just like the standard normal
 It looks exactly like the standard normal, except with larger
tails.
 The values of T require a parameter, degrees of freedom, to find
the value on the table
 Degrees of freedom are equal to n – 1
 As n increases, t approaches Z
Example
 Construct a 95% confidence interval for the mean weight of
apples (in grams):
 x-bar = 183, s = 14.1, n = 16
 First find t
 Alpha/2 = .025 ( because of 95% confidence)
 df = n-1 = 15
 t = 2.13
 183±2.13*14.1/sqrt(16)
 (175.5 and 190.5)
 We are 95% confident that the mean weight of apples is
between 175.5 and 190.5 grams