M ARIO F . T RIOLA

1. STATISTICS ELEMENTARY Chapter 6 Estimates and Sample Sizes MARIO F. TRIOLA EIGHTH EDITION

2. 6-2 Estimating a Population Mean: Large Samples 6-3 Estimating a Population Mean: Small Samples 6-4 Sample Size Required to Estimate µ 6-5 Estimating a Population Proportion Chapter 6Estimates and Sample Sizes

3. Estimating a Population Mean: Large Samples 6-2

4. n > 30 The sample must have more than 30 values. Simple Random Sample All samples of the same size have an equal chance of being selected. Assumptions Data collected carelessly can be absolutely worthless, even if the sample is quite large.

5. Estimator a formula or process for using sample data to estimate a population parameter Estimate a specific value or range of values used to approximate some population parameter Point Estimate a single value (or point) used to approximate a population parameter Definitions The sample mean x is the best point estimate of the population mean µ.

6. Confidence Interval a range (or interval) of values used to estimate the true value of the population parameter Lower # < population parameter < Upper # Definition EXAMPLE: 97.9ºF <  < 99.2ºF

7. the probability 1 -  which represents the relative frequency of times the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times This value is often expressed in percent form, usually 90%, 95%, or 99%  = 10%  = 5%  = 1% Definition Degree of Confidence • (or level of confidence or confidence coefficient)

8. Correct: We are 95% confident that the interval from 98.08to 98.32actuallydoes contain the true valueof . (This means that if we were to select many different samplesof size 106 and construct the confidence intervals,95% of themwould actually contain the value of the population mean .) Wrong: There is a 95% chance that the true value of will fallbetween 98.08 and 98.32. Interpreting a Confidence Interval 98.08o <µ < 98.32o

9. Confidence Intervals from 20 Different Samples Figure 6-1

10. The number separating sample statistics that are likely to occur from those unlikely to occur. Definition Critical Value The Critical Value z2 2 2 LIKELY LIKELY z2 -z2 z=0 UNLIKELY

11. Finding z2for 95% Degree of Confidence 95%  = 5% = .05 2 = .025 .95 .025 .025 1.96 -1.96 Use invNorm to find the cutoff score for an area (probability) of 0.025. z2 -z2 Critical Values z2 = 1.96

12. Margin of Error Definition • the maximum likely difference observed between sample mean x and true population mean µ. • denoted by E

13. Margin of Error Definition For a given confidence level, the maximum likely difference observed between sample mean x and true population mean µ. Denoted by E. distribution of sample means for samples of size n E E µ x - E x x + E

14. Margin of Error Definition For a given confidence level, the maximum likely difference observed between sample mean x and true population mean µ. Denoted by E. distribution of sample means for samples of size n E E x - E µ x + E x x -E < µ < x +E upper limit lower limit

15. Confidence Interval (or Interval Estimate) for Population Mean µ(Based on Large Samples: n >30) x - E < µ < x + E µ = x + E (x - E, x + E)

16. Constructing a Confidence Interval for µ using the TI-83 (Based on a Large Sample: n > 30 ) • Press Stat, select Tests, select 7:Zinterval. • To use DATA IN A LIST, select Data. • Enter σ (since n>30, use s if n not known), • enter list name, enter 1 for frequency, and enter confidence level. • Then select calculate and press enter. • Round to one more decimal place than original data. • To use SUMMARY STATISTICS, select Stats. • Enter σ (or s if necessary), x, n, and confidence level. • Then select calculate and press enter. • Round to same number of decimal places as • sample mean.

17. n = 106 x = 98.20o s = 0.62o Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the 95% confidence interval for the population mean μ. Press Stat, select Tests, 7:Zinterval. We’re givenSUMMARY STATISTICS, so select Stats. Enter .62 for σ, 98.2 for x, 106 for n, and .95 for confidence level. Select calculate and press enter. The calculator returns (98.082, 98.318). Round to (98.08, 98.32). If we were to select many different samples of the same size, 95% of the confidence intervals would actually contain the population mean μ.

18. Margin of Error E = z/2 •  n Calculating E When  Is Unknown • If n> 30, we can replace  in the formula above by the sample standard deviation s. • If n 30, the population must have a normal distribution and we must know  to use the formula.

19. Finding E from a Confidence Interval E = (upper conf. interval limit) - (lower conf. interval limit) 2 Finding a Point Estimate of μ from a Confidence Interval x = (upper conf. interval limit) + (lower conf. interval limit) 2

20. Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the 95% confidence interval for the population mean μand find the margin of error E. Press Stat, select Tests, 7:Zinterval. We’re givenSUMMARY STATISTICS, so select Stats. Enter .62 for σ, 98.2 for x, 106 for n, and .95 for confidence level. Select calculate and press enter. n = 106 x = 98.20o s = 0.62o Answer rounds to (98.08, 98.32).