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The probabilities will be the number of observations with a quality divided by the number of observations. So, for a), how many women with breast cancer tested positive? How many women had breast cancer?
3a would be 7/8 in percent form and 3b would be 70/992 in percent form.
Of course, these are our estimates for those probabilities because we only have a sample of 1000 people; not the entire population of women (probably within some age range).
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4.) is an important question as well. Clearly we can see that a woman is much more likely to test positive for breast cancer if she has it than if she doesn't, but we can also describe this numerically. To test for independence we assert a null hypothesis:
H0: All people test positive at the same rate regardless of their actual cancer status
The probability of observing at least 7 out of 8 people with breast cancer test positive when the population mean is 7% would be:
8*(0.07)^7*(0.93) + (0.07)^8 = 0.0000000618
This is far below any threshold we would reasonably imagine someone might still entertain the null hypothesis so we can safely reject it.
Can you see where I produced this expression above?
so that would be for 4
Yeah. Next, how many people out of the 1000 tested positive?