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Statistics Practice Problems with Solutions

1. In a 2003 study, the Accreditation Council for Graduate Medical Education found that medical residents’ mean number of hours worked in a week is μ = 81 hours. Suppose the number of hours per week by medical residents is approximately normally distributed with a standard deviation of σ = 6.9 hours.

What is the probability that a randomly selected medical resident works less than 70 hours in a week?

What is the probability that a randomly selected medical resident works more than 100 hours in a week?

2. The Wechsler Intelligence Scale for Children is approximately normally distributed with a mean µ = 100 and a standard deviation σ = 15.

What proportion of children will score above 125?

What proportion of children will score below 90?

What intelligence score will place a child in the top 5% of all children?

3. The average male in an industrialized country lives µ = 70 and σ = 6.3. Use this information to answer the following 4 questions.

What percentage of males lives longer than 75 years?

What percentage of males lives between 65 and 75 years?

What percentage of males lives less than 65 years?

What percentage of males lives between 55 and 60 years?

4. In a distribution of scores with a mean of 1500 and a standard deviation of 250, what raw score corresponds with the top 25%?

4. The following three questions refer to a distribution with µ = 60 and σ = 4.3

The raw score corresponding to a z-score of 0.00 is _____________

The raw score corresponding to a z-score of -1.51 is _____________

The z-score corresponding to a raw score of 68.7 is _____________

6. Men in third-world countries have life expectancies of µ = 60 and σ = 4.3.

Men in industrialized countries have a life expectancy of µ = 70 and σ = 6.3. If a man in a third-world country lives to be 62 and a man in an industrialized country lives to be 75, who lived longer relative to their age distribution? Make sure to provide evidence for your answer using the standard normal distribution.

7. In a distribution with a mean of 50 and a standard deviation of 5:

What z-score cuts off the top 10% of this (or any) distribution?

What raw score cuts off the top 10% of this distribution?

8. When flipping a coin, heads and tails are mutually exclusive because:

1. if the coin comes up heads, it cannot also come up tails.

2. if the coin comes up heads on one toss, it has no influence on whether the coin comes up heads or tails on the next toss.

3. sampling is with replacement.

4. sampling is without replacement.

9. Jake is having a party for all of his friends in his apartment complex. He knows they all have very different tastes, so he stocks his refrigerator with a large selection. Jake has 12 bottles of Corona beer, 24 bottles of Shock Top beer, 24 bottles of Heineken beer, 8 bottles of wine coolers, and 12 bottles of Coke.

Billy wants any beer. What is the probability that the first beverage Jake randomly grabs is a beer?

10.What is the probability of drawing an ace out of a standard deck of 52 cards?

11.What is the probability of drawing a red card out of a standard deck of 52 cards?

12.What is the probability of drawing a red ace out of a standard deck of 52 cards?

13.A letter of the English alphabet is chosen at random. Find the probability that the letter selected:

is a vowel (consider y a consonant) _____________

is any letter which follows p in the alphabet _____________

14.If I flip a coin 5 times which set of heads (H) and tails (T) outcomes is more likely:

1. HHHHH

2. TTTTT

3. HTHTH

4. all are equally likely

15.There are 105 applicants for a job with a new coffee shop. Some of the applicants have worked at coffee shops before and some have not served coffee before. Some of the applicants can work full-time, and some can only work part-time. The exact breakdown of applicants is as follows:

Coffee Shop Experience (E) No Coffee Shop Experience (not E) Total

Available Full-Time (F) 20 12 32

Available Part-Time (not F) 42 31 73

Total 62 43 105

Find each of the following probabilities.

P(E): The probability someone has coffee shop experience.

P(F): The probability someone is available full-time.

P(not E): The probability someone has no coffee shop experience.

P(E and F): The probability someone has coffee shop experience and is available full-time.

16.Earthquakes: The magnitude of earthquakes since 1900 that measure 0.1 or higher on the Richter scale in California is approximately normally distributed, with μ = 6.2 and σ = 0.5, according to data obtained from the Geological Survey.

What is the probability that a randomly selected earthquake in California has a magnitude greater than 6.0?

What is the probability that a randomly selected earthquake in California has a magnitude less than 6.4?

What is the probability that a randomly selected earthquake in California has a magnitude between 5.8 and 7.1?

The great San Francisco Earthquake of 1906 had a magnitude of 8.25. Is an earthquake of this magnitude unusual in California? What is the probability that a randomly selected earthquake has a magnitude greater than 8.25?

17.Gestation Period: The length of human pregnancies is approximately normally distributed with mean μ = 266 days and standard deviation σ = 16 days.

What is the probability a randomly selected pregnancy lasts less than 260 days? What is the probability that a random sample of 25 pregnancies has a mean gestation period of less than 260 days?

What is the probability that a random sample of 49 pregnancies has a mean gestation period of more than 260 days?

18.Oil Change: The shape of the distribution of the time required to get an oil change at a 10-minute oil-change facility is unknown. However, records indicate that the mean time for an oil change is 11.4 minutes and the standard deviation for oil-change time is 3.2 minutes.

What is the probability that a random sample of 36 oil changes results in a sample mean time less than 10 minutes?

19.Alzheimer’s: A patient recently diagnosed with Alzheimer’s disease takes a cognitive abilities test. The patient scores a 45 on the test. The cognitive abilities test is approximately normally distributed with μ= 52 and σ = 5. What is the probability that a randomly selected patient with Alzheimer’s will score less than 45 on the test?

20.Parkinson’s: Another patient with Parkinson’s disease takes the same cognitive abilities test as in question 1 above and scores a 54. The cognitive abilities test is approximately normally distributed with μ= 52 and σ = 5. What is the probability that a randomly selected patient with Parkinson’s will score more than 54 on the test?

21.Fifth Grade: A fifth grader takes a standardized achievement test and scores a 148. The test is approximately normally distributed with μ= 125 and σ = 15. What is the probability that a randomly selected fifth grader will score less than 148 on this test?

22.Spatial Abilities: Pat and Chris took a spatial abilities test that is approximately normally distributed with μ= 80 and σ = 8. Pat scored a 76 and Chris scored a 94. What is the probability that a randomly selected individual will score between 76 and 94?

23.Principals’ Salaries: According to the National Survey of Salaries and Wages in Public Schools, the mean salary paid to public high school principals in 2004-2005 was μ = $71,401. Assume the distribution is normally distributed with a standard deviation σ = $26,145. What is the probability that a random sample of 100 public high school principals has an average salary under $65,000?

24.Teaching Supplies: According to the National Education Association, public school teachers spend an average μ = $443 of their own money each year to meet the needs of their students. Assume the distribution is normally distributed with a standard deviation σ = $175. What is the probability that a random sample of 50 public school teachers spends an average of more than $400 each year to meet the needs of their students?

25.Energy Needs during Pregnancy: Suppose the total energy need during pregnancy is normally distributed with mean μ = 2600 kcal/day and standard deviation σ = 50 kcal/day. What is the probability that a randomly selected pregnant woman has an energy need of more than 2625 kcal/day? Is this result unusual? What is the probability that a random sample of 20 pregnant women has a mean energy need of more than 2625 kcal/day? Is this result unusual?