STATISTICS

see attached for questions

Continuos Probability Distributions

1.The results of a common standardized test used in psychology research is designed so that the population mean is 145 and the standard deviation is 10. A subject earns a score of 120. What is the

z-score for this raw score?
z-score = 

2.

Adult men have heights with a mean of 69.0 inches and a standard deviation of 2.8 inches. Find the height of a man with a z-score of -1.3 (to 2 decimal places)

3. Which of the following is false? Hint: It might be useful to sketch the distributions.

· Calculating percentiles based on the Z table is only appropriate for observations that come from a nearly normal distribution.

· The Z score for the median of a left skewed distribution is negative.

· Z scores are defined for observations from distributions of any shape and skew.

· The Z score for the median of a symmetric distribution is approximately 0.

4.The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 55 ounces and a standard deviation of 6 ounces.
Use the Standard Deviation Rule, also known as the Empirical Rule.
Suggestion: sketch the distribution in order to answer these questions.
a) 68% of the widget weights lie between

oz and  oz

b) What percentage of the widget weights lie between 43 and 61 ounces?  %
c) What percentage of the widget weights lie above 37 ?  %

5.The time to complete an exam is approximately Normal with a mean of 40 minutes and a standard deviation of 4 minutes.

The bell curve below represents the distribution for testing times. The scale on the horizontal axis is equal to the standard deviation. Fill in the indicated boxes.

6.The results of a common standardized test used in psychology research is designed so that the population mean is 155 and the standard deviation is 40. A subject earns a score of 187. What is the z-score for this raw score?
z-score = 

7.The time it takes me to wash the dishes is uniformly distributed between 6 minutes and 15 minutes.
What is the probability that washing dishes tonight will take me between 8 and 13 minutes?
Give your answer accurate to two decimal places.

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https://s3.amazonaws.com/myopenmath/cfiles/2408/ReadingE6

https://s3.amazonaws.com/myopenmath/cfiles/2408/VocabularyC56

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Normal Distribution

1.A manufacturer knows that their items have a normally distributed lifespan, with a mean of 4.8 years, and standard deviation of 0.8 years.
If you randomly purchase one item, what is the probability it will last longer than 3 years?
Round answer to three decimal places

2.In the country of United States of Heightlandia, the height measurements of ten-year-old children are approximately normally distributed with a mean of 54.5 inches, and standard deviation of 2.6 inches.
What is the probability that the height of a randomly chosen child is between 51.1 and 51.4 inches? Do not round until you get your your final answer, and then round to 3 decimal places.
Answer= (Round your answer to 3 decimal places.)

3.The amounts of nicotine in a certain brand of cigarette are normally distributed with a mean of 0.921 g and a standard deviation of 0.329 g. Find the probability of randomly selecting a cigarette with 0.395 g of nicotine or less.
P(X < 0.395 g) =  Enter your answer as a number accurate to 4 decimal places. NOTE: Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

4.Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find P75, the 75-percentile. This is the temperature reading separating the bottom 75% from the top 25%.
P75 = °C
(Round answer to three decimal places)

5.Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 6.7-in and a standard deviation of 1.2-in. Due to financial constraints, the helmets will be designed to fit all men except those with head breadths that are in the smallest 4% or largest 4%.
What is the minimum head breadth that will fit the clientele?
min = 
What is the maximum head breadth that will fit the clientele?
min = 
Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

6.The mean daily production of a herd of cows is assumed to be normally distributed with a mean of 40 liters, and standard deviation of 12.6 liters.
A) What is the probability that daily production is between 56 and 57.9 liters? Do not round until you get your your final answer.
Answer= (Round your answer to 4 decimal places.)
Warning: Do not use the Z Normal Tables…they may not be accurate enough since WAMAP may look for more accuracy than comes from the table.

7.The mean daily production of a herd of cows is assumed to be normally distributed with a mean of 30 liters, and standard deviation of 5.8 liters.
A) What is the probability that daily production is less than 46.6 liters?
Answer= (Round your answer to 4 decimal places.)
B) What is the probability that daily production is more than 26.2 liters?
Answer= (Round your answer to 4 decimal places.)
Warning: Do not use the Z Normal Tables…they may not be accurate enough since WAMAP may look for more accuracy than comes from the table.

8.A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 183.4-cm and a standard deviation of 1.9-cm.
Find the probability that the length of a randomly selected steel rod is less than 182.1-cm.
P(X < 182.1-cm) =  Enter your answer as a number accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

9.Company XYZ know that replacement times for the DVD players it produces are normally distributed with a mean of 7.4 years and a standard deviation of 1.7 years.
If the company wants to provide a warranty so that only 4.1% of the DVD players will be replaced before the warranty expires, what is the time length of the warranty?
warranty =  years
Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

10.Sketch the region corresponding to the statement P(z>−0.9)P(z>-0.9)

Shade: . Click and drag the arrows to adjust the values.

-1.5

Sketch the region corresponding to the statement P(−c

Shade: . Click and drag the arrows to adjust the values.
 

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http://onlinestatbook.com/2/normal_distribution/history_normal.html

http://onlinestatbook.com/2/normal_distribution/areas_normal.html

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The Central Limit Theorem

1.A simple random sample of size n=40 is obtained from a population with a mean of 20 and a standard deviation of 5. Is the sampling distribution normally distributed? Why?

· Yes, the sampling distribution is normally distributed because the sample size is greater than 30.

· Yes, the sampling distribution is normally distributed because the population is normally distributed.

· No, the sampling distribution is not normally distributed because the population is not normally distributed.

· No, the sampling distribution is not normally distributed because the population mean is less than 30.

2.A population has parameters μ=69.3μ=69.3 and σ=3.6σ=3.6. You intend to draw a random sample of size n=233n=233.
What is the mean of the distribution of sample means?
μ¯x=μx¯= 
What is the standard deviation of the distribution of sample means?
(Report answer accurate to 2 decimal places.)
σ¯x=σx¯= 

3.A population of values has a normal distribution with μ=28.6μ=28.6 and σ=60.6σ=60.6. You intend to draw a random sample of size n=13n=13.
Find the probability that a single randomly selected value is greater than 23.6.
P(X > 23.6) = 
Find the probability that a sample of size n=13n=13 is randomly selected with a mean greater than 23.6.
P(M > 23.6) = 
Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

4.A manufacturer knows that their items have a lengths that are skewed right, with a mean of 19.2 inches, and standard deviation of 3.8 inches.
If 49 items are chosen at random, what is the probability that their mean length is greater than 18.6 inches?
(Round answer to four decimal places)

5.A particular fruit’s weights are normally distributed, with a mean of 423 grams and a standard deviation of 35 grams.
If you pick 25 fruits at random, then 18% of the time, their mean weight will be greater than how many grams?
Give your answer to the nearest gram.

6.A population of values has a normal distribution with μ=12μ=12 and σ=49.3σ=49.3. You intend to draw a random sample of size n=215n=215.
What is the mean of the distribution of sample means?
μ¯x=μx¯= 
What is the standard deviation of the distribution of sample means?
(Report answer accurate to 2 decimal places.)
σ¯x=σx¯= 

7.A population of values has a normal distribution with μ=106.5μ=106.5 and σ=88.2σ=88.2. You intend to draw a random sample of size n=46n=46.
Find the probability that a single randomly selected value is greater than 113.
P(X > 113) = 
Find the probability that a sample of size n=46n=46 is randomly selected with a mean greater than 113.
P(M > 113) = 
Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

8.The manager of a computer retails store is concerned that his suppliers have been giving him laptop computers with lower than average quality. His research shows that replacement times for the model laptop of concern are normally distributed with a mean of 4.2 years and a standard deviation of 0.4 years. He then randomly selects records on 43 laptops sold in the past and finds that the mean replacement time is 4.1 years.
Assuming that the laptop replacement times have a mean of 4.2 years and a standard deviation of 0.4 years, find the probability that 43 randomly selected laptops will have a mean replacement time of 4.1 years or less.
P(M < 4.1 years) =  Enter your answer as a number accurate to 4 decimal places. NOTE: Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted. Based on the result above, does it appear that the computer store has been given laptops of lower than average quality?

· Yes. The probability of this data is unlikely to have occurred by chance alone.

· No. The probability of obtaining this data is high enough to have been a chance occurrence.

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https://s3.amazonaws.com/myopenmath/cfiles/2408/ReadingE8

http://onlinestatbook.com/2/sampling_distributions/samp_dist_mean.html

http://onlinestatbook.com/2/sampling_distributions/samp_dist_meanM.html

http://onlinestatbook.com/2/calculators/normal_dist.html

http://onlinestatbook.com/2/calculators/inverse_normal_dist.html

OTHER QUESTIONS

1.A group of people were asked if they had run a red light in the last year. 478 responded “yes”, and 438 responded “no”.
Find the probability that if a person is chosen at random, they have run a red light in the last year.
  
Give your answer as a fraction or decimal to 4 decimal places

2.

Probability

Scores

0.1

0

0.05

2

0.2

3

0.25

9

0.35

11

0.05

14

Find the expected value of the above random variable.   

3.The table summarizes results from 986 pedestrian deaths that were caused by automobile accidents.

Driver
Intoxicated?

Pedestrian Intoxicated?

Yes

No

Yes

61

81

No

267

577

If one of the pedestrian deaths is randomly selected, find the probability that the pedestrian was intoxicated or the driver was intoxicated.
Report the answer as a percent rounded to one decimal place accuracy. You need not enter the “%” symbol.
prob = %

4.About 9% of the population has a particular genetic mutation. 1000 people are randomly selected.
Find the standard deviation for the number of people with the genetic mutation in such groups of 1000. Round your answer to three decimal places

5.Assume that a procedure yields a binomial distribution with a trial repeated n=5n=5 times. Use some form of technology like Excel or StatDisk to find the probability distribution given the probability p=0.432p=0.432 of success on a single trial.
(Report answers accurate to 4 decimal places.)

k

P(X = k)

0

1

2

3

4

5

6.The amounts of nicotine in a certain brand of cigarette are normally distributed with a mean of 0.976 g and a standard deviation of 0.308 g. Find the probability of randomly selecting a cigarette with 0.452 g of nicotine or less.
P(X < 0.452 g) =  Enter your answer as a number accurate to 4 decimal places. NOTE: Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

7.Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 5.6-in and a standard deviation of 0.9-in. Due to financial constraints, the helmets will be designed to fit all men except those with head breadths that are in the smallest 1.5% or largest 1.5%.
What is the minimum head breadth that will fit the clientele?
min = 
What is the maximum head breadth that will fit the clientele?
min = 
Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

8.The mean daily production of a herd of cows is assumed to be normally distributed with a mean of 31 liters, and standard deviation of 9.4 liters.
A) What is the probability that daily production is less than 3.2 liters?
Answer= (Round your answer to 4 decimal places.)
B) What is the probability that daily production is more than 35.3 liters?
Answer= (Round your answer to 4 decimal places.)
Warning: Do not use the Z Normal Tables…they may not be accurate enough since WAMAP may look for more accuracy than comes from the table.

9.Company XYZ know that replacement times for the quartz time pieces it produces are normally distributed with a mean of 15.4 years and a standard deviation of 2.1 years.
If the company wants to provide a warranty so that only 1.5% of the quartz time pieces will be replaced before the warranty expires, what is the time length of the warranty?
warranty =  years
Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

10.A population has parameters μ=27.8μ=27.8 and σ=91.5σ=91.5. You intend to draw a random sample of size n=221n=221.
What is the mean of the distribution of sample means?
μ¯x=μx¯= 
What is the standard deviation of the distribution of sample means?
(Report answer accurate to 2 decimal places.)
σ¯x=σx¯= 

11. The manager of a computer retails store is concerned that his suppliers have been giving him laptop computers with lower than average quality. His research shows that replacement times for the model laptop of concern are normally distributed with a mean of 3.5 years and a standard deviation of 0.6 years. He then randomly selects records on 25 laptops sold in the past and finds that the mean replacement time is 3.1 years.
Assuming that the laptop replacement times have a mean of 3.5 years and a standard deviation of 0.6 years, find the probability that 25 randomly selected laptops will have a mean replacement time of 3.1 years or less.
P(M < 3.1 years) =  Enter your answer as a number accurate to 4 decimal places. NOTE: Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted. Based on the result above, does it appear that the computer store has been given laptops of lower than average quality?

· No. The probability of obtaining this data is high enough to have been a chance occurrence.
· Yes. The probability of this data is unlikely to have occurred by chance alone.