Practice Midterm1. In a survey of a very large population, the frequencies of two health problems are to beestimated from a single simple random sample. It is thought that one of the health problemseffects about 3% of the population and the effects about 40%. Ignore the finite populationcorrection factor in answering the questions.[a] How large should the sample b e in order for the standard errors for both estimates to beless that 0.01?[b] What are the actual standard errors for this sample size?2. In Stat 21, Spring 94 there were 314 students, and 131 of these students were female. Ina simple random sample of 91 of the 314 STAT 21 students, let¯V be the fraction of the 91respondents who were female and played a video game in the week prior to the survey. Toestimate π, the fraction of women in the class that played video games in the week prior tothe survey, we use the estimator:ˆπ =¯V131/314.Prove that ˆπ is an unbiased estimator of π.3. Breast cancer is one of the most common malignancies among women in the U.S. If itis detected early enough – before the cancer spreads – chances of successful treatment aremuch better. Do screening programs speed up detection enough to matter?The first large-scale trial was run by the Health Insurance Plan of Greater New York,starting in 1963. The subjects (all members of the plan) were 62,000 women age 40 to 64.These women were divided at random into 2 equal groups. In the treatment group, womenwere encouraged to come in for annual screening, including examination by a doctor andx-rays. About 20,200 women in the group did come in for screening; but 10,800 refused.The control group was offered usual health care. All the women were followed for manyyears. Results for the first 5 years are shown in the table below.Cause of DeathBreast Cancer All OtherNumber Rate Number RateTreatment GroupExamined 20,200 23 1.1 428 21Refused 10,800 16 1.5 409 38Total 31,000 39 1.3 837 27Control Group 31,000 63 2.0 879 28[a] Does screening save lives? Which numbers prove your point?[b] The death rate from all causes among women who refused screening is about double thedeath rate among women who came in for the exam. Did screening cut the death rate in1half?4. In a famous sociological study in the 1950s, 784 high school students were asked whichtwo of ten given attributes were most desirable in their fathers. The following table showshow the desirability of the attribute “being a college graduate” was rated by male and femalestudents. Did the males and females value this attribute differently? Conduct a statisticaltest to answer this question.Male FemaleMentioned 86 55Not mentioned 283 360Provide each of the following:1. Null Hypothesis:2. Alternative hypothesis:3. Test statistic:4. p-value:5. Is the result statistically significant? Explain6. What are your