show all calculations thanks 10. Each student in a large statistics class of 240

Question

show all calculations thanks

10. Each student in a large statistics class of 240 students is asked to independently toss a fair coin 100 times and to record the number of heads and tails obtained (a) Each student is asked to independently construct a 95% confidence interval for the true probability of heads. Assume that each student constructs the confi- dence interval correctly. Which of the following statements is correct regarding the constructed confidence intervals by the whole class A. None of the confidence intervals will contain 0.5 since the students con structed the confidence intervals independently. B. We would expect approximately 12 of the confidence intervals to contain 0.5. C. All of the confidence intervals will contain 0.5 since the students constructed the confidence intervals independently. D. We would expect approximately 228 of the confidence intervals to contain the 0.5 (b) Suppose a random sample of 6 students is selected from the class. Each student is then asked to independently test the hypothesis that heads and tails are equally likely at the 5% significance level. What is probability that less than half the tests in this sample resulted in a type I error.

Explanation / Answer

95 confidence interval for proportion interpretation -

: For about 95% of all randomly selected samples from the population, the confidence interval computed in this manner captures the population proportion.

hence 240* 0.95 = 228 sample will contain 0.5

hence

option D) is correct

b)

type i error = 0.05

we reject the null hypothesis when it is true

n = 6

X = number of students who make type i error

X follow binomial distribution with n = 6 , p = 0.05

P(X < 3) = P(X = 0) + P(X =1) +P(X = 2)

= 0.99777

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