Confidence Intervals

1
Unit 14

• Confidence Intervals
• Means

2
Estimating Population Means

• Youve learned how to create confidence intervals
  for proportions.
• Now its time to do the same thing for means.
• Recall the format for the confidence interval for
  proportions

3
Recall the CLT

• The z-score in the CI formula reflected the fact
  that if n is large, sample proportions are
  normally distributed.
• Sample means are normally distributed (provided
  ) with a standard deviation equal to

4
This Seems Reasonable

• By direct comparison if the formula for
  proportions is
• Then the formula for means might well be

5
Why Cant Life Be Easy?

• Two problems
• Sample means are only normally distributed if the
  sample size is 30 or more
• We dont know the population standard deviation,
  , but rather and estimate, namely the sample
  standard deviation, s.

6
Gossets t

• Working in the late 1800s Gosset had a job
  that was the envy of many. He was the quality
  control manager for Guinness Brewery in Dublin,
  Ireland. Gosset used small samples of size 3 or 4
  since larger samples might be unhealthy. He
  noticed that his estimates were off considerably
  and this bugged him. He took time off and got a
  graduate degree in statistics!

7
The Rest of the Story

• He figured out when he used in his
  calculation the sampling distribution for the
  mean was no longer normal. He figured out what
  the new model should be and called it the
  t-distribution.
• The Guinness Company didnt give Gosset a lot of
  support for his work. In fact, they had a policy
  against publishing results. So Gosset published
  his results using the name Student and the model
  is now called Students t distribution.

8
Students t-distribution

• The shape of the t-distribution changes with
  different sample sizes.
• There is a whole family of t-distributions
  whose shape depends on the sample size
• As the sample size gets larger the Student
  t-distribution is approximately the same as the
  standard normal distribution.
• Interactive C-6, Basics Basics2

9
T-Values

• In practice, nowadays, everyone uses either a
  calculator or computer to determine t-values.
• What is important to remember is that, unlike
  z-scores, t-values depend on the sample size
• When the sample size is small, t-values are large

10
Now We Have It

• To evaluate a confidence interval for a
  population mean we use the formula
• The margin of error is given by the term

11
Margin of Error

• To decrease the margin of error you could
• Increase the sample size
• Reduce (if possible) the standard deviation
• Decrease the t-value by reducing your confidence
  level

12
Teachers Salaries

• A random sample of 288 Nevada teachers produced
  the following 90 confidence interval
• Explain what this means
• What is the margin of error for this confidence
  interval?
• Would the margin of error be smaller or larger
  for a 95 confidence interval?

13
Crawling

• Data collected by child development scientists
  produced the following 95 CI for the average age
  (in weeks) at which babies first begin to crawl.
• Explain clearly what this means
• What is the margin of error? Interpret.
• Would a 90 CI be wider or narrower?

14
On the TI-83/84

• On the TI 83/84
• In the STAT TESTS menu choose 8TInterval
• Use can choose between data entered in a list or
  you can enter the sample mean, standard deviation
  and sample size
• Enter the desired confidence level
• Notice you dont have to enter a t-value, the
  calculator takes care of that for you!

15
Body Temperature

• Researchers measured the body temperature of 52
  randomly selected adults and recorded the mean
  body temperature to be 98.285 with a sample
  standard deviation of .6824.
• Find both 95 and 99 CI for the average adult
  body temperature.

16
Solution

• Using the TI-83 we get
• 95 CI
• 99 CI

17
Hot Dogs

• A nutrition laboratory tests 40 reduced sodium
  hot dogs, finding that the mean sodium content is
  310 mg, with a standard deviation of 36 mg
• Find a 95 confidence interval for the mean
  sodium content of this brand of hot dog.
• Clearly explain what your interval means

18
Solution

• The 95 interval is
• With 95 confidence we can state that the mean
  sodium content of this brand of hot dog is
  between 298.49 mg and 321.51 mg.

19
Yogurt

• Consumer Reports tests 14 brands of vanilla
  yogurt and found the following number of calories
  per serving
• 160 200 220 230 120 180 140
• 130 170 190 80 120 100 170
• Create a 95 CI for the average calorie content
  of vanilla yogurt
• Labeling indicates the calorie content to be 120
  calories er serving. Comment on this.

20
On the TI - 83

• First enter the data into a list on your TI-83
• Form the T Interval menu select Input
  Data
• Next to List enter the list that contains your
  data
• Leave Freq1 alone
• Enter your C-Level

21
Solution

• The 95 CI is
• The data indicates that the average number of
  calories per serving for vanilla yogurt is
  between 132 and 184 calories per serving.
• Since 120 calories is outside (and below 132), it
  appears the average calories per serving is more
  than what is stated on the label.