STAT 250 Dr Kari Lock Morgan Inference for

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STAT 250 Dr. Kari Lock Morgan Inference for Proportions Chapter 6. 1, 6. 2,

STAT 250 Dr. Kari Lock Morgan Inference for Proportions Chapter 6. 1, 6. 2, 6. 3, 6. 7, 6. 8, 6. 9 • Formulas for standard errors • Normal based inference Statistics: Unlocking the Power of Data Lock 5

Confidence Interval Formula IF SAMPLE SIZES ARE LARGE… From N(0, 1) From original data

Confidence Interval Formula IF SAMPLE SIZES ARE LARGE… From N(0, 1) From original data Statistics: Unlocking the Power of Data From bootstrap distribution Lock 5

Formula for p-values IF SAMPLE SIZES ARE LARGE… From original data From randomization distribution

Formula for p-values IF SAMPLE SIZES ARE LARGE… From original data From randomization distribution Statistics: Unlocking the Power of Data From H 0 Compare z to N(0, 1) for p-value Lock 5

Standard Error • Wouldn’t it be nice if we could compute the standard error

Standard Error • Wouldn’t it be nice if we could compute the standard error without doing thousands of simulations? • We can!!! Statistics: Unlocking the Power of Data Lock 5

Standard Error Formulas Parameter Distribution Proportion Standard Error Normal Difference in Proportions Normal Mean

Standard Error Formulas Parameter Distribution Proportion Standard Error Normal Difference in Proportions Normal Mean t, df = n – 1 Difference in Means t, df = min(n 1, n 2) – 1 Correlation t, df = n – 2 Statistics: Unlocking the Power of Data Lock 5

SE Formula Observations �n is always in the denominator (larger sample size gives smaller

SE Formula Observations �n is always in the denominator (larger sample size gives smaller standard error) �Standard error related to square root of 1/n �Standard error formulas use population parameters… (uh oh!) �For intervals, plug in the sample statistic(s) as your best guess at the parameter(s) �For testing, plug in the null value for the parameter(s), because you want the distribution assuming H 0 true Statistics: Unlocking the Power of Data Lock 5

Hormone Replacement Therapy �Until 2002, hormone replacement therapy (HRT), estrogen and/or progesterone, was commonly

Hormone Replacement Therapy �Until 2002, hormone replacement therapy (HRT), estrogen and/or progesterone, was commonly prescribed to post-menopausal women. This changed in 2002, when the results of a large clinical trial were published � 8506 women were randomized to take HRT, 8102 were randomized to placebo. 166 HRT and 124 placebo women developed invasive breast cancer �Does hormone replacement therapy cause increased risk of breast cancer? Statistics: Unlocking the Power of Data Lock 5

Interval for Proportion �First: What are the chances a woman not taking HRT develops

Interval for Proportion �First: What are the chances a woman not taking HRT develops invasive breast cancer? �Give a 90% confidence interval. �What is the sample statistic? �What is z* for a 90% interval? �What is the standard error? Statistics: Unlocking the Power of Data Lock 5

Sample Statistic � 8506 women were randomized to take HRT, 8102 were randomized to

Sample Statistic � 8506 women were randomized to take HRT, 8102 were randomized to placebo. 166 HRT and 124 placebo women developed invasive breast cancer Statistics: Unlocking the Power of Data Lock 5

z* Statistics: Unlocking the Power of Data Lock 5

z* Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

Interval for Proportion Statistics: Unlocking the Power of Data Lock 5

Interval for Proportion Statistics: Unlocking the Power of Data Lock 5

Bootstrap Interval Statistics: Unlocking the Power of Data Lock 5

Bootstrap Interval Statistics: Unlocking the Power of Data Lock 5

Interpretation “We are 90% confident that the true proportion of post-menopausal women not taking

Interpretation “We are 90% confident that the true proportion of post-menopausal women not taking HRT who develop invasive breast cancer is between 0. 013 and 0. 018” Do you believe this sentence is accurate? a) Yes b) No Statistics: Unlocking the Power of Data Lock 5

Your Turn! � 8506 women were randomized to take HRT, 8102 were randomized to

Your Turn! � 8506 women were randomized to take HRT, 8102 were randomized to placebo. 166 HRT and 124 placebo women developed invasive breast cancer �Give a 95% confidence interval for the proportion of women taking HRT who develop invasive breast cancer: a) b) c) d) (0. 013, 0. 027) (0. 015, 0. 025) (0. 017, 0. 023) (0. 019, 0. 021) Statistics: Unlocking the Power of Data Lock 5

Testing �When testing, use the null value for p, rather than the sample statistic

Testing �When testing, use the null value for p, rather than the sample statistic �So, if testing whether the proportion of people not taking HRT who develop invasive breast cancer is less than 0. 1, the standard error would be Statistics: Unlocking the Power of Data Lock 5

Hormone Replacement Therapy � 8506 women were randomized to take HRT, 8102 were randomized

Hormone Replacement Therapy � 8506 women were randomized to take HRT, 8102 were randomized to placebo. 166 HRT and 124 placebo women developed invasive breast cancer �Does hormone replacement therapy cause increased risk of breast cancer? Statistics: Unlocking the Power of Data Lock 5

Hypothesis Test �State hypotheses. �Calculate z-statistic. �Compare to standard normal distribution to find p-value.

Hypothesis Test �State hypotheses. �Calculate z-statistic. �Compare to standard normal distribution to find p-value. �Make a conclusion. Statistics: Unlocking the Power of Data Lock 5

Hypotheses Does hormone replacement therapy cause increased risk of invasive breast cancer? p 1

Hypotheses Does hormone replacement therapy cause increased risk of invasive breast cancer? p 1 = proportion of women taking HRT who get invasive breast cancer p 2 = proportion of women not taking HRT who get invasive breast cancer a) H 0: p 1= p 2, Ha: p 1≠ p 2 H 0: p 1= p 2, Ha: p 1> p 2 c) H 0: p 1= p 2, Ha: p 1< p 2 b) Statistics: Unlocking the Power of Data Lock 5

Calculate z-statistic null value = 0 What to use for p 1 and p

Calculate z-statistic null value = 0 What to use for p 1 and p 2? ? ? Have to assume null is true… Statistics: Unlocking the Power of Data Lock 5

Null Values �Testing a difference in proportions: H 0: p 1 = p 2

Null Values �Testing a difference in proportions: H 0: p 1 = p 2 �Use the overall sample proportion from both groups (called the pooled proportion) as an estimate for both p 1 and p 2 �Note that this is in between the sample proportions for each group. Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

z-statistic Statistics: Unlocking the Power of Data Lock 5

z-statistic Statistics: Unlocking the Power of Data Lock 5

p-value Statistics: Unlocking the Power of Data Lock 5

p-value Statistics: Unlocking the Power of Data Lock 5

p-value Statistics: Unlocking the Power of Data Lock 5

p-value Statistics: Unlocking the Power of Data Lock 5

Conclusion Does this provide evidence that hormone replacement therapy increases risk of invasive breast

Conclusion Does this provide evidence that hormone replacement therapy increases risk of invasive breast cancer? a) Yes b) No Statistics: Unlocking the Power of Data Lock 5

Your Turn! �Same trial, different variable of interest. � 8506 women were randomized to

Your Turn! �Same trial, different variable of interest. � 8506 women were randomized to take HRT, 8102 were randomized to placebo. 502 HRT and 458 placebo women developed any kind of cancer. �Does hormone replacement therapy cause increased risk of cancer in general? a) Yes b) No Statistics: Unlocking the Power of Data Lock 5

Margin of Error For a single proportion, what is the margin of error? a)

Margin of Error For a single proportion, what is the margin of error? a) b) c) Statistics: Unlocking the Power of Data Lock 5

Margin of Error You can choose your sample size in advance, depending on your

Margin of Error You can choose your sample size in advance, depending on your desired margin of error! Given this formula for margin of error, solve for n. Statistics: Unlocking the Power of Data Lock 5

Margin of Error Statistics: Unlocking the Power of Data Lock 5

Margin of Error Statistics: Unlocking the Power of Data Lock 5

Margin of Error Suppose we want to estimate a proportion with a margin of

Margin of Error Suppose we want to estimate a proportion with a margin of error of 0. 03 with 95% confidence. How large a sample size do we need? (a) About 100 (b) About 500 (c) About 1000 (d) About 5000 Statistics: Unlocking the Power of Data Lock 5

To Do �Read Sections 6. 1, 6. 2, 6. 3, 6. 7, 6. 8,

To Do �Read Sections 6. 1, 6. 2, 6. 3, 6. 7, 6. 8, 6. 9 �Do HW 6 a (due Friday, 4/10) Statistics: Unlocking the Power of Data Lock 5