The average time to run the 5K fun run is 24 minutes and the standard deviation is 2.2 minutes. 48 runners are randomly selected to run the 5K fun

1.  

The average time to run the 5K fun run is 24 minutes and the standard deviation is 2.2 minutes. 48 runners are randomly selected to run the 5K fun run. Round all answers to 4 decimal places where possible and assume a normal distribution.

a. What is the distribution of XX? XX ~ N(,)

b. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)

c. What is the distribution of ∑x∑x? ∑x∑x ~ N(,)

d. If one randomly selected runner is timed, find the probability that this runner's time will be between 24.1237 and 24.5237 minutes. 

e. For the 48 runners, find the probability that their average time is between 24.1237 and 24.5237 minutes. 

2.  

Each sweat shop worker at a computer factory can put together 5 computers per hour on average with a standard deviation of 0.8 computers. 50 workers are randomly selected to work the next shift at the factory. Round all answers to 4 decimal places where possible and assume a normal distribution.

a. What is the distribution of XX? XX ~ N(,)

b. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)

c. What is the distribution of ∑x∑x? ∑x∑x ~ N(,)

d. If one randomly selected worker is observed, find the probability that this worker will put together between 4.9 and 5 computers per hour. 

e. For the 50 workers, find the probability that their average number of computers put together per hour is between 4.9 and 5. 

f. Find the probability that a 50 person shift will put together between 240 and 245 computers per hour. 

g. For part e) and f), is the assumption of normal necessary? YesNo

h. A sticker that says "Great Dedication" will be given to the groups of 50 workers who have the top 10% productivity. What is the least total number of computers produced by a group that receives a sticker? computers per hour (round to the nearest computer)

i. Find the probability that the randomly selected 48 person team will have a total time more than 1132.8. 

j. For part e) and f), is the assumption of normal necessary? NoYes

k. The top 15% of all 48 person team relay races will compete in the championship round. These are the 15% lowest times. What is the longest total time that a relay team can have and still make it to the championship round? minutes.

3. Suppose that the amount of time that students spend studying in the library in one sitting is normally distributed with mean 46 minutes and standard deviation 21 minutes. A researcher observed 6 students who entered the library to study. Round all answers to 4 decimal places where possible.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. What is the distribution of ∑x∑x? ∑x∑x ~ N(,)
4. If one randomly selected student is timed, find the probability that this student's time will be between 33 and 40 minutes. 
5. For the 6 students, find the probability that their average time studying is between 33 and 40 minutes. 
6. Find the probability that the randomly selected 6 students will have a total study time more than 324 minutes. 
7. For part e) and f), is the assumption of normal necessary? NoYes
8. The top 10% of the total study time for groups of 6 students will be given a sticker that says "Great dedication". What is the least total time that a group can study and still receive a sticker? minutes

4.  

The average amount of money that people spend at Don Mcalds fast food place is $7.1300 with a standard deviation of $1.7300. 11 customers are randomly selected. Please answer the following questions, and round all answers to 4 decimal places where possible and assume a normal distribution.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. What is the distribution of ∑x∑x? ∑x∑x ~ N(,)
4. What is the probability that one randomly selected customer will spend more than $6.3976? 
5. For the 11 customers, find the probability that their average spent is more than $6.3976. 
6. Find the probability that the randomly selected 11 customers will spend less than $70.3736. 
7. For part e) and f), is the assumption of normal necessary? YesNo
8. The owner of Don Mcalds gives a coupon for a free sundae to the 3% of all groups of 11 people who spend the most money. At least how much must a group of 11 spend in total to get the free sundae? $

5.  

The average score for games played in the NFL is 20.7 and the standard deviation is 9.2 points. 42 games are randomly selected. Round all answers to 4 decimal places where possible and assume a normal distribution.

1. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
2. What is the distribution of ∑x∑x? ∑x∑x ~ N(,)
3. P(¯xx¯ < 20.3706) = 
4. Find the 68th percentile for the mean score for this sample size. 
5. P(20.6706 < ¯xx¯ < 22.4098) = 
6. Q3 for the ¯xx¯ distribution = 
7. P(∑x∑x < 914.3652) = 
8. For part c) and e), is the assumption of normal necessary? Yes or No