1. Author(s): Brenda Gunderson, Ph.D., 2011
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3. What parameter are we learning how to estimate and test hypotheses about now? (CLICKER in your answer) m

4. Recall our results about Sample Means page 109

5. More on the Standard Deviation of
Standard deviation of the sample mean:
Interpret: approximately the average distance of the possible values (for repeated samples of same size n) from the true population mean m.
Note: sample size increases  standard deviation decreases.
Problem: in practice we would not know the value of s.

6. The Standard Error of Standard error of the sample mean:
Interpret: Estimates, approximately, the average distance of the possible values (for repeated samples of same size n) from the true population mean m.

7. 9.9 Standardized Statistics for Means pg 112
z-statistic for a sample mean:
z = has (approximately) a standard normal distribution N(0,1).
Dilemma =

8. 9.9 Standardized Statistics for Means
Replace s with s …
won’t be approximately N(0,1) instead it has a ….

9. Student’s t-Distribution
About t-distributions ...
Symmetric, unimodal, centered at 0
Flatter with heavier tails compared to the N(0,1)
As df increases … t distribution  N(0,1)
Can still use ideas about standard scores.
Tables A.2 & A.3 summarize percentiles for t-distributions

10. 9.10 Sampling Distribution for any Statistic page 113
The sampling distribution of a statistic … is the distribution of possible values of the statistic for repeated samples of same size from a population.
Every statistic has a sampling distribution, but the appropriate distribution may not always be normal, or even be approximately bell-shaped.
Example 9.19 (pg ) shows good example.

11. 11.1 Intro to CIs for Means page 115
Interval = a range of reasonable values for the parameter with an associated high level of confidence. “We are 95% confident that …”
95% confidence level describes our confidence in the procedure we used to make the interval. If we repeated the procedure many times, we would expect about 95% of the intervals to contain the population parameter.

12. 11.2 CI Module 3: CI for a Population Mean m
Design of a highway sign. Q: What is the mean maximum distance at which drivers are able to read the sign?
Data: Researcher will take a r.s. of n = 16 drivers and measure the maximum distances (in feet) at which each can read the sign.
Population parameter: m = _______________ mean maximum distance to read the sign for _________________
Sample estimate: = _______________ mean maximum
distance to read the sign for __________________________
But we know the sample estimate may not equal m, in fact, the possible values vary from sample to sample.

13. Recall Sampling Distribution for Mean pg 116
If is sample mean for a random sample of size n from a normal model, then the distribution of the sample mean is:
Central Limit Theorem
If is the sample mean from a random sample of size n
from a population with any model, with mean m,
and standard deviation s, then when n is large, the
sampling distribution of the sample mean is approximately .

14. The Standard Error of the Sample Mean
Estimates, roughly, the average distance of the possible values from the true population proportion m.
One-sample t Confidence Interval for m
where t* is from a t(n – 1) distribution.
Interval requires have a r.s. from normal popul. If sample size large (n > 30), assumption of normality not so crucial and result is approx.

15. Try It! Using Table A.2 page 117
(a) Find t* for a 90% CI based on n = 12 obs.
(b) Find t* for a 95% CI based on n = 30 obs.
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition
Used with permission.

16. Try It! Using Table A.2 (c) Find t* for a 95% CI based on n = 54 obs.
(d) What happens to t* as n gets larger?
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition.
2012.
Used with permission.

17. Try It! CI for Mean Maximum Distance pg 119
r.s. of n = 16 drivers; measured max distance (in ft) at which can read sign.
a. Verify conditions. Told sample was a random sample so need to check if normal model for ‘max distance’ for the population is reasonable.

18. CI for Mean Maximum Distance: Check conditions

19. CI for Mean Maximum Distance: Std Error
r.s. of n = 15 drivers; max distance (in ft)
b. Compute sample mean max distance and std error.

20. CI for Mean Maximum Distance: Interpret
c. Use 95% CI to estim the mean maximum distance at which all drivers can read the sign. Write a paragraph that interprets this interval and the confidence level.

21. CI for Mean Maximum Distance: SPSS output

22. Q3: Environmental Proposal – Consider the following table that summarizes the attitude (in favor, indifferent, or opposed) to a particular environmental proposal and the political party for the 100 U.S. senators.
In Favor
Indifferent
Opposed
Total
Democrat 27 15 18 60
Republican 13 10 17 40 25 35
100
a. What is the probability that a randomly selected Democrat will be in favor of the proposal?

23. Q3: Environmental Proposal
In Favor
Indifferent
Opposed
Total
Democrat 27 15 18 60
Republican 13 10 17 40 25 35
100
a. What is the probability that a randomly selected Democrat will be in favor of the proposal? b. To determine whether being a Democrat was independent from being in favor, you would compare answer to part (a) with another probability. What is it? Hence, being a Democrat and being in favor are: dependent independent

24. Q4: Satisfied with your first-year experience
Q4: Satisfied with your first-year experience? – Ugrads today are less satisfied with first year college experience than 5 years ago. Survey based on sample of 500 undergraduates completing first year. Fifty-eight percent reported they were happy, down from 63 percent (established rate 5 years prior). Is result sufficient evidence to conclude ‘satisfied’ rate is significantly lower than the prior rate of 63%? Use a 5% significance level
a. Following defin of parameter of interest is not complete, so you are asked to correct it. Let p = the population proportion of all first year college students

25. Q4: Satisfied with your first-year experience
Q4: Satisfied with your first-year experience? – Is result sufficient evidence to conclude ‘satisfied’ rate is significantly lower than the prior rate of 63%? Use a 5% significance level.
b. State the appropriate hypotheses. H0: ________________________ Ha:_________________________

26. Q4: Satisfied with your first-year experience
Q4: Satisfied with your first-year experience? – Is result sufficient evidence to conclude ‘satisfied’ rate is significantly lower than the prior rate of 63%? Use a 5% significance level.
c. Test statistic value was z = Find p-value. There (circle one) is is not sufficient evidence to conclude that the satisfied rate is significantly lower than prior rate of 63%.