The time it takes waiting to ride a popular roller coaster at an amusement park is normally distributed with a mean of 24 minutes and a standard deviation of 7.5 minutes.

Question

1. What is the probability that a random rider waits more than a half of an hour? Provide a Sketch w/ shading, proper notation, and probability statement. 
2. What is the probability that 5 random riders' average wait time is more than half of an hour? Be sure to include all 4 steps neatly: 1)Assumptions/Conditions 2)Sampling distribution (sketch) 3) Probability calculation 4) Interpretation of results
3. Explain thoroughly in context of the problem what the difference is between the probabilities calculated in (1) versus (2)

Answer 1

1) Prob ( x>=30) = 1 - Prob(x<=30)

= 1 - Prob ( z <= (30-24)/7.5)

= 1 - Prob ( z <= 0.8)

= 1 - 0.788144601

= 0.211855399

< 21.2%

2) hypothesis test given normal distribution

hypothetical mean = 30

standard deviation = 7.5

assuming 95% confidence ---> z_level alpha = 1.96

(30-24)/ D where D = 7.5/ sqrt(5) = 3.3541019662496845446137605030969....

1.788854382- < 1.96 so it passes the hypothesis test

Prob ( z < 1.788) = 0.963184539 < 96.32%