Content by Week Week of October 14 27

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1 Content by Week Week of October Learning objectives By the end of this week, you should be able to: Understand the purpose and interpretation of confidence intervals for the mean, Calculate confidence intervals for means using the t distribution, Understand when confidence intervals for means and proportions have the coverage claimed. Topics The t distribution and how it varies with the degrees of freedom Confidence interval for a mean Relation of role of independence, sample size, and the shape of the probability distribution of the data on the coverage of confidence intervals for proportions and means

2 A new probability distribution: The (student s) t distribution Features: Symmetric Bell shaped Centred at 0

3 A new probability distribution: The (student s) t distribution Features: Symmetric, Bell shaped, Centered at 0 How does it differ from the Normal distribution? o Heavier tails N(0,1) t with df=

4 A new probability distribution: The (student s) t distribution Features: Symmetric, Bell shaped, Centred at 0 How does it differ from the Normal distribution? o Heavier tails N(0,1) t with df=

5 A new probability distribution: The (student s) t distribution Features: Symmetric, Bell shaped, Centred at 0 How does it differ from the Normal distribution? Heavier tails. How heavy? Varies with the degrees of freedom N(0,1) t with df=25 t with df=5 t with df=1 The smaller the degrees of freedom, the heavier the tails. As the degrees of freedom grows, the t distribution approaches the Normal distribution

6 The consequence of heavy tails: N(0,1) t with df=25 t with df=5 t with df= Put in order from largest to smallest: A. P(X>3) where X has a t distribution with 1 degree of freedom B. P(Y>3) where Y has a t distribution with 5 degrees of freedom C. P(Z>3) where Z has a N(0,1) distribution

7 The consequence of heavy tails: N(0,1) t with df=25 t with df=5 t with df=1 α/2 t α/2 t α/2 α/ Put in order from largest to smallest: A. t α/2 from a t distribution with 1 degree of freedom B. t α/2 from a t distribution with 5 degrees of freedom C. z α/2 from a N(0,1) distribution

8 t table: Since it needs to include various degrees of freedom (df), the table only gives a few probability values, and only the positive quantiles. Also note that the probability is in the margins and the quantiles are in the cells of the table.

9 t table: What happens as the number of degrees of freedom gets large? On an exam, if you need 66 df, use 60. Why is this the conservative choice?

10 Suppose X is a random variable with a t distribution with 4 degrees of freedom. What is P(X < 2.13)? What is the value of a such that P(X >a) = 0.01?

11 Suppose X is a random variable with a t distribution with 4 degrees of freedom. What is P(X < 2.13)? 0.05 What is the value of a such that P(X >a) = 0.01? 3.75

12 Confidence interval for a mean: Want to estimate μ ( population mean or true mean or mean from model in the theoretical world ). We don t know μ. Estimate it with the mean (or average) calculated from the data (the sample mean ). We have a numerical value for this. The confidence interval gives a range of plausible values for what μ could be, based on our observed data.

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18 What is the effect on the confidence interval of quadrupling n?

19 TRUE or FALSE: If the TRUE variance of the sampling distribution was known, then it would be appropriate to use a t distribution to construct confidence intervals for the mean.

20 From OpenIntro Statistics exercise 4.8 A question on the General Social Survey is For how many days during the past 30 days was your mental health, which includes stress, depression, and problems with emotions, not good? Based on responses from 1,151 US residents, the survey reported a 95% confidence interval of 3.40 to 4.24 days in a) What numbers did they need to calculate this confidence interval? b) What does a 95% confidence level mean in this context? c) If a new survey asking the same questions was to be done with 500 Americans, would the standard error of the estimate be larger, smaller, or about the same. Assume the standard deviation has remained constant since 2010.