Central Limit Theorem and Confidence Intervals

• Testing Hypothesis
• When do we trust our data?
• Are the differences real?
• Are the data reliable?
• Plus or minus what?

• mean: x bar vs. mu
• standard deviation: sd vs. sigma

• Computer failures
• Date rejections
• Grading
• Reaction Time
• How many green balls are in the urn

• Get a sample
• Is one observation sufficient?
• How about three?
• How about a thousand?
• More is better
• Bigger sample gives us a better idea of what is going on
• One "test" could be a fluke

• Yes
• Which ones?
• Ones with greater variance

• The distribution of the means of samples
• If you did this exp again, would you find the same thing?
• This might take a bit to get your head around.

• The normal distribution can help.
• If you survey N people, your survey will get some mean response X₁.
• If you took another survey of N people from the same population, this survey would have a mean X₂.
• If you took a bunch of surveys and plotted the means on a histogram, you would find something that looked like a normal distribution.
• Even if the data you are sampling is not normally distributed.
• Sampling Distribution of the Mean.

• As the size of the samples increases, the distribution of the means becomes normal.
• But what is the standard deviation of these sample means?

• The size of the sample
• The standard deviation of the population measure

• This distribution of survey results would follow a normal distribution.
• Mean of the sampling distribution of the mean = x bar
• Standard deviation of the sampling distribution of the mean = sigma/(N)^1/2
• Underlying distribution
• mu = 100
• sigma = 10

• As N (sample size) increases, the variability in this distribution decreases substantially.
• By N = 1000, the true mean is quite likely to be very close to the mean obtained in the survey.

• The sampling distribution of the mean
• Mean = x bar = mu (with big N)
• standard deviation = sigma/(N)^1/2
• As N gets larger, the standard deviation of the sampling distribution gets smaller.
• Diminishing returns for additional observations.
• A cost-benefit analysis

• Sampling distributions
• The central limit theorem
• Creating confidence intervals
• Using confidence intervals

• How is the sample mean related to the population mean?
• Requires thinking about sampling distributions.
• Sampling distribution of the mean.
• Distribution of means of samples of a particular size.

• If we take a lot of samples of size N
• Look at the mean of these samples
• What will the mean of these means be?
• Approximately the population mean
• How closely this approximates the population mean will depend on the sample size.
• The larger the sample, the more accurate is the estimate of the mean.

• Tells us something neat about means
• For any population
• For large sample sizes, the sampling distribution of the mean is normally distributed.
• Even if the population values are not normally distributed.

• The sampling distribution is very useful.
• Generally, we want to know the population mean.
• How can we make a guess about what that mean actually is?
• Collect a sample.
• The mean of that sample is an estimate of the population mean.

• Where is the population mean likely to be?
• You can construct an interval that is likely to contain the population mean.
• This interval is called a confidence interval.
• You can build an interval that is as wide as the confidence you want to have in it.

• Suppose you want to know the average number of raisins in a box of Raisin Bran.
• Kelloggs has said that the there is some variability in the number of raisins in a box.
• They list the standard deviation as 26 raisins.
• This is the population standard deviation.
• Take a sample of 100 boxes and count them.
• Mean = 263
• How confident do you want to be that the actual mean falls in the interval?
• 50%, 95%, 99%?  Pick one.

• A procedure to use when sigma is known
• For the confidence interval
• Use the mean of the sample (263)
• Use the standard deviation of the mean
• 26/(100)1/2 = 2.6
• Find the z-score associated with that level of confidence
• z(95%)=1.96  (leaving 2.5% in each tail)
• 95% confidence interval (z * 2.6)
• 258 < mean < 268

• To construct a confidence interval when the population standard deviation is known
• Collect a sample
• Find the mean
• Find the z-score for the desired confidence
• Leave p/2 of the probability in each tail
• The confidence interval
• Mean ± (z * standard deviation of mean)

What is x bar?

--- 50

What is the standard deviation of the mean?

--- 10/4^(1/2)= 5

What is the 95% confidence interval?

--- z(95%) = 1.96

--- margin = 1.96*5 = plus or minus 9.8

Are the pop-o-matic people wrong?

--- Null Hypotheses: kernels equals 15
--- Alternative Hypothesis: kernels not equal to 15

--- Calculate z score: (15-50)/5=-7

--- Conclusion: 7 z-scores from the mean is about a zero chance. More likely aliens will interrupt class.





More examples

1. Jane needs to score in the top 2.5% percent of an aptitude test to qualify for a job. The mean of test is 75 and standard deviation is 10. How high of a score does she need to get?

2. The standard deviation for the number of students attending a class is 5. Over five classes, 25, 35, 30, 26, and 20 students attend the class. What is the mean number of students attending for these five classes? What is the standard deviation associated with this mean?

3. Mia needs to figure out the mean number of hours children spend watching TV per week. She needs the standard deviation associated with the mean to be less than or equal to 2. The standard deviation for how many hours children watch TV is 6 hours per week. How many children does she need to survey?

4. Dave runs an experiment with 1241 subjects. Kate runs the same experiment (on the same population) with 476 subjects. What is Kate's standard deviation of the mean divided by Dave's standard deviation of the mean?

5. The weight of a seal has a standard deviation of 20 pounds. An animal trainer at Sea World is interested in how heavy the seals are at the park (on average). Eight seals at the park are weighed and weigh 200, 220, 300, 175, 218, 315, 180, and 200 pounds.

What is the mean weight of the seals?
What is the confidence 95% confidence interval for the above mean?
What is the confidence 99% confidence interval for the above mean?