Question

1. A friend who works in a big city owns two cars, one small and one large. One-quarter of the time he drives the small car to work, and three-quarters of the time he takes the large car. If he takes the small car, he usually has little trouble parking and so is at work on time with probability 0.8. If he takes the large car, he is on time to work with probability 0.6. Given that he was at work on time on a particular morning, what is the probability that he drove the large car? (Give the answer to three decimal places.)

2. A friend who works in a big city owns two cars, one small and one large. One-quarter of the time he drives the small car to work, and three-quarters of the time he takes the large car. If he takes the small car, he usually has little trouble parking and so is at work on time with probability 0.8. If he takes the large car, he is on time to work with probability 0.6. Given that he was at work on time on a particular morning, what is the probability that he drove the large car? (Give the answer to three decimal places.)

3. The amount of money spent by a customer at a discount store has a mean of $100 and a standard deviation of $30. What is the probability that a randomly selected group of 50 shoppers will spend a total of more than $5600? (Hint: The total will be more than $5600 when the sample average exceeds what value?) (Round the answer to four decimal places.)
P(total > 5600)= ___

4. The two intervals (114.1, 115.5) and (113.8, 115.8) are confidence intervals (computed using the same sample data) for μ = true average resonance frequency (in hertz) for all tennis rackets of a certain type.

(a) What is the value of the sample mean resonance frequency? (Hint: Where is the confidence interval centered?)  

please answer all the parts thanks

Answer 1

1.

P(Small) = 1/4

P(Big) = 3/4

P(On time | Small) = 0.8

P(On time | Big) = 0.6

By law of total probability,

P(On time) = P(Small) P(On time | Small) + P(Big) P(On time | Big)

= (1/4) * 0.8 + (3/4) * 0.6 = 0.65

By Bayes theorem,

P(Big | On time) = P(On time | Big) P(Big) / P(On time)

= 0.6 * (3/4) / 0.65

= 0.692

2.

Repeated question (Same as question 1)

3.

The total will be more than $5600 when the sample average exceeds 5600 / 50 = 112

Standard error of mean = = 30 / = 4.24

P(total > 5600)= P( > 112) = P[Z > (112 - 100)/4.24] = P[Z > 2.83]

= 0.002

4.

sample mean resonance frequency = (Lower limit + Upper Limit) / 2

= (114.1 + 115.5) / 2

= 114.8