Day 15: STA205 - Introduction to Statistical Methods

• Mean: 74.9
• SD: 12.1
• Check that I added up your problem totals correctly.

• Important: I'll need to take the exams back -- or at least the take-home part -- to do some test assessment -- they'll be returned to you for good on Thursday.

• Some comments:
  • Problem 1: avoid confusing population in its "traditional sense" with population in its "statistical sense": the population is the number of years worked for each and every employee of the huge company.
  • Problem 4:
    • The probability that xbar is <= 240 should be very small, right? Since the mean for the population is 265, and n=850.
    • So xbar=240 should be very unlikely, coming from a population that eats 265 eggs a year: they almost certainly eat fewer eggs....
  • Take home:
    • We want to know how well the empirical rule is working, so we compare it to actual data.
    • We certainly didn't want you using a normal distribution (e.g. calculate z-values) for this distribution, which is far -- very far! -- from normal.
    • Ask yourself: why did they want me to sort the data?

1. estimate , with given confidence (section 6.1);
2. find out what the sample size should be to estimate , with given precision (section 6.2).

• Estimating with given confidence (Section 6.1)
  • What does it mean to estimate something "with confidence"?

  • "confidence interval" -- what should that mean?

  • z-interval for (see p. 214)

  • Here's a table of common choices for z:

    Desired Confidence	90%	95%	99%	???
    z-value	1.645	1.960	2.576	get it from the z-table!

  • See p. 221 for conditions under which we may do this.

  • Example (follow Strategy of p. 216):
    • #1, p. 221
    • What if we needed only a 90% confidence interval?
    • What if we wanted a 99% confidence interval?

• On to Objective 2 (Section 6.2):
  • What if we require a 95% confidence interval with a known precision?

    That is, we want to know that lies within a certain interval with 95% confidence.

  • Then we will have to design our experiment well! In particular we will have to select an appropriate sample size.

  • We know that the larger the sample size, the "tighter" the normal curve of the distribution of .

  • Hence n determines how tight the curve needs to be, by the equation , where the value of z corresponds to the desired confidence (in this case, 95%), and E corresponds to the "half-width" of the interval.

  • Example:
    • #4, p. 227
    • Notice that it's not necessary to know the mean value in order to calculate the sample size! Only the margin of error E, z, and the known value of
    • From the course website of worked examples

Please Note: this handout was created for a different class, so chapter and page references are not valid.

1. Null hypothesis:
2. Alternative hypothesis:
3. Decision rule: , then reject the null in favor of the alternative.
4. Test statistic: z-statistic,
5. Compute p-value (value as extreme as Z=-2.08 or more so): , hence reject the null in favor of the alternative.

6. Interpretation of the results: older folks apparantly don't eat as many eggs as other adults (our estimate is 240 eggs per year).

• #1, p. 242 (proportion)
• From the course website of worked examples