Question

Assume that 20% of U.S. adults subscribe to the "five-second rule." That is, they would eat a piece of food that fell onto the kitchen floor if it was picked up within five seconds.

(1) Assuming this is a binomial situation, calculate the probability that more than 325 respondents in a survey of 1500 people, would subscribe to the five-second rule.

(Round your answer to the nearest 3 decimal places, show what you typed into the calculator,and define n, p, and r.)

(2) Provide evidence that conducting a normal approximation in this scenario would be appropriate.

(3) The survey went out to 1500 people . Calculate the mean and standard deviation.

(Label the mean and standard deviation clearly. Round to 3 decimal places where necessary.)

(4) Use the normal approximation method to answer the same question asked in part calculate the probability that more than 325 respondents in a survey of 1500 people would subscribe to the five-second rule. Show your work and round your answer to the nearest 3 decimal places.

(5) Your answers to questions 1 and 4 should have been fairly similar. Explain why your work in question 2 ensures that the probabilities you calculated using both methods would be close to one another.

Answer 1

Probability of more than 325 respondents following five second rule is the one which is selected above ie 0.05097.

The above probability cannot be directly calculated as the computation directly involves too much time to get the answer. So we seek the help of the calculator. But a direct method can be employed if certain assumptions are true. ie if np and n(1-p) should be greater than 10. Let us see whether the assumptions are satisfied...

np=1500*0.2=300>10 and n(1-p)=1500*0.8=1200>10>Hence by the central limit theorem since n=1500 is quite large number the respondents following 5 second rule can be approximated to normal distribution with mean=300 and variance=240; S.D=15.492. Let X denotes the number of respondents following five second rule

Then

It can be proved easily that binomial tends to normal for large n. Here n>1500 ( quite a large number). So normal distribution can be used to calculate binomial cumulative probabilities.