1)in a city located by the equator, the average yearly temperature will exceed 100 degrees Fahrenheit 62% of the time. what is the probability that for the past 50 years, the temperature has exceeded 100 degrees at most 57% of the time.

2)a 2014 study of 300 randomly selected residents of a city found that 7% of them had been exposed to the mosquito born virus that causes dengue fever. suppose the actual percentage of people in the city who have been exposed to the virus is 6%. the researcher of the study believe the discrepancy is due to the sample size, so they decided to conduct another survey with double the sample size. what is the probability that the proportion of residents from this new survey that has been exposed to the virus will be even higher than the previous study?

3)it is known that 80% of former students in Mr. Tsai's classes regularly celebrate An-Tsai commercialism day every year. suppose we want to determine the probability that from a random sample chosen of former students, less than 75% celebrate An-Tsai commercialism day. what is the smallest sample size necessary to fulfill the criteria of normality before solving the problem using sampling distributions?

Question)

 

Q1)

 

The probability can be calculated by using the normal distribution approximation

 

Z = (p - p₀)/SQRT(p₀*(1-p₀)/N)

 

Where, 

 

p is the observed proportion = 0.62

 

p₀ is the hypothesized proportion = 0.57

 

N is the sample size = 50

 

Z = (0.57 - 0.62)/SQRT(0.62*0.38/50) = -0.7284

 

P (Temperatures greater than 100⁰F <= 57%) = P (Z <= -0.7284) = 0.2332

 

Q2)

 

Z = (p - p₀)/SQRT(p₀*(1-p₀)/N)

 

N will increase to 600 from 300 in the earlier study

 

We need to find the probability that the proportion of exposed residents in the new survey is greater than 7%

 

Z = (0.07 - 0.06)/SQRT(0.06*0.94/600) = 1.0314

 

P (proportion of exposed residents in new survey > 7%) = P (Z > 1.0314) = 0.1512

 

Q3)

 

To fulfill the criteria of normality N*p and N*(1-p) must be greater than 5

 

In this question, the value of p = 0.80, which is the proportion of students in Mr. Tsai's class who celebrate the day

 

N*p > 5 

 

N*0.8 > 5

N*(4/5) > 5 

N > 25/4 = 6.25 ------------------- (1)

 

N*(1-p) > 5

 

N*0.2 > 5

N*(1/5) > 5 

N > 25 -----------------------(2)

 

Using conditions (1) & (2), we see that N > 25

 

Therefore, the minimum value of N to fulfill the criteria is 26.

 

If you have any doubts, please comment below. I'll be happy to resolve them. 

 

Step-by-step explanation

Question)

 

Q1)

 

P (Temperatures greater than 100⁰F <= 57%) = P (Z <= -0.7284) = 0.2332

 

Q2)

 

P (proportion of exposed residents in new survey > 7%) = P (Z > 1.0314) = 0.1512

 

Q3)

 

To fulfill the criteria of normality N*p and N*(1-p) must be greater than 5

 

Therefore, the minimum value of N to fulfill the criteria is 26.