Statistical Sciences 2244A/B Study Guide - Final Guide: Binomial Distribution, Sampling Distribution, Statistical Hypothesis Testing
22 May 2018
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Department
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Stats 2244
Two sample Inference
Difference in proportions
Scenario
- Risk of breast cancer is thought to be influenced by events between age of menarche and age at
first child birth. An international study was conducted to evaluate whether the risk of breast
cancer differs for young versus old mothers. A SRS of 3099 women who were <30 y old when
they had their first child was identified; of these women, 1890 were hospitalized for breast
cancer. A SRS of 7546 women who were 30+ when they had their first child was identified. Of
these women, 7038 were hospitalized for breast cancer. All others had no history
Approximate z test for difference in p
- So this scenario deals with 2 samples: the sample of young mothers and the sample of older
mothers
- The data should be summarized with proportions bc breast cancer history is qualitative
- We use an approx. z test for a difference in proportion
- We look at null and alternate hypothesis
o H0: p1=p2
o The alternative hypothesis could be left tailed, right tailed or two tailed
- The test statistic we are dealing with is an approx. z test
- We have 2 p hat values: proportions of successes (women with breast cancer in our two
samples)
- Then we have our hypothesized value for our parameter
o Here, our parameter is the difference in proportions (p1=p2)
- In the denominator, we have the SE of the relevant sampling distribution
o Here, bc the parameter we are looking at is the difference in proportions, and the
statistic we are working with is the difference in sample proportions
o So the denominator is the estimates SD for the difference in sample proportions
- P bar is the pooled estimate of the proportion
o We are pooling (bringing together) all the successes in our two samples
o x1 and x2 are the count of number of successes in sample 1 and sample 2
o q bar is calculated using the complement (1-p bar)
- conditions:
o samples are independent (not matched pairs) →to evaluate this, we have to look at the
sampling strategy
o both samples are SRS
o using normal as an approx. for an underlying binomial distribution
▪ we need to binomial conditions to apply
▪ if we are dealing with proportions and some success vs failure, we have 2
outcomes, fixed sample sizes, and if the rest of the conditions are valid, then the
binomial setting should apply
▪ we need the number of successes and failtures in both sample 1 and separately
in sample 2 to be at least 5
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Document Summary
Risk of breast cancer is thought to be influenced by events between age of menarche and age at first child birth. An international study was conducted to evaluate whether the risk of breast cancer differs for (cid:494)young(cid:495) versus (cid:494)old(cid:495) mothers. A srs of 3099 women who were <30 y old when they had their first child was identified; of these women, 1890 were hospitalized for breast cancer. A srs of 7546 women who were 30+ when they had their first child was identified. Of these women, 7038 were hospitalized for breast cancer. So this scenario deals with 2 samples: the sample of young mothers and the sample of older mothers. The data should be summarized with proportions bc breast cancer history is qualitative. We use an approx. z test for a difference in proportion. We look at null and alternate hypothesis: h0: p1=p2, the alternative hypothesis could be left tailed, right tailed or two tailed.
