Sept 3 2018

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1.

Show the outcomes, the number of combinations, and the probabilities for the discrete random variable for the number of tails in three successive flips of a coin. Also calculate the mean and the variance.

2.

Baseball’s World Series follows a best-of-seven series. The series winner must win four games to be crowned World Champion. However, teams do not always play seven games. As soon as one team wins four games, the series is over. The following table lists the number of games played in every World Series from 1905 to 2011, along with the probability of that happening. For example, 20 times (19.4%) the series was over in four games. The probability table is given below.

  • Calculate the expected number of games, the variance, and the standard deviation for the number of games played in a world series.
  • Networks bid for World Series broadcasting rights, yet they cannot know with certainty how many games they will broadcast. Assume they don’t make a profit unless at least six games are played in the series. What is the probability they will make a profit in any given year?

3.

The binomial tables used by the author are based on a cumulative distribution. Given a sample size (n) and an overall probability of success (p), you can solve probability problems for the binomial distribution using the tables. It is important to learn to use the tables because they help solve some problems much faster than using the binomial formula, which may require many calculations. The key to understanding the tables is that the table is based on a cumulative distribution. Sometimes you need to perform addition and subtraction to get the correct answer. Answer the following questions using the Binomial Table for n = 25 and p = 0.4.

  • Solve the probability for x = 8 using the binomial formula.
  • Next, get the same answer from the binomial tables. To use the table you have to subtract the cumulative probability for x = 7 from the cumulative probability for x = 8. This leaves the exact probability of 8. Confirm that the table calculation equals the formula calculation (to at least four decimal places).
  • What is the probability of more than 8, P(x > 8)? To solve this, you need to subtract the table value for 8 from 1.0000.
  • What is the probability of 4 or less, P(x < 5)?
  • What is the probability of between 6 and 9, P(x > 5 and x < 10)?

4.

Supposed an estimated 30% of high school students smoke cigarettes. You randomly select 25 high school students and survey them about their attitudes on smoking, and you ask them if they smoke. Look at the probabilities associated with the number of smokers out of 25.This is a binomial probability distribution problem with n = 25 and p = 0.30.

  • What is the probability that at least 8 students smoke? P(x ≥ 8)?
  • What is the probability that exactly 5 students smoke? P(x = 5)?
  • Calculate the mean, variance, and standard deviation for this problem.

5.

The standard normal table used by the author is based on μ = 0 and σ = 1. The table shows the probability from the center of the distribution out so many standard deviations in the right side of the curve. The right side of the distribution has a total probability of 0.5 (as does the left side). Given any mean and standard deviation from a normal distribution, computing a z-score converts the distribution to a standard normal. Since the normal distribution is symmetrical, the left side is a mirror image of the right side, and you can use the absolute value of a negative z-score to find probabilities on the left side of the curve. It is important to learn to use the standard normal table to solve normal distribution problems.

To get started, find the probabilities based on a normal distribution with a mean of 100 and a standard deviation of 10; that is, μ = 100 and σ = 10.

  • The probability between 100 and 115
  • The probability greater than 115
  • The probability between 120 and 125
  • The probability less than 175
  • The probability greater than 181

6.

The height of females aged 20 to 29 follows a normal distribution with μ = 64.1 and σ = 2.8.

  • What is the probability that a female aged 20 to 29 will be under 60 inches tall?
  • What is the probability that a female aged 20 to 29 is taller than 66 inches?
  • What is the probability that a female aged 20 to 29 will be between 61 and 68 inches tall?
  • What is the probability that a female aged 20 to 29 will be shorter than 68 inches?
  • Modeling agencies like models 5’10” or taller. What is the probability a female aged 20 to 29 is greater or equal to 70 inches? Is this a rare event? Explain your answer.

7.

The length of pregnancies follows a normal distribution, with μ = 268 days and σ = 15 days.

  • What is the probability that a pregnancy will last less than 280 days?
  • What is the probability that a pregnancy will last less than 250 days?
  • Pregnancy is considered “at term” when gestation attains 37 complete weeks (259 days) but is less than 42 complete weeks (294 days). What proportion of pregnancies fall within this period?
  • What is the number of days at the 50th percentile?
  • New medical technology and procedures put viability as low as 24 weeks (168 days), but a child born that early only has a 50% chance of survival. Chances of survival are much better at 32 weeks. What is the probability that a pregnancy will last less than 32 weeks (224 days)? Is this a rare event? Explain your answer.

8.

The intelligence quotient (IQ) test can be scored based on a normal distribution with a mean of 100 and a standard deviation of 15: IQ ~ N(100, 15).

  • What proportion of the population has an IQ that falls between 90 and 120?
  • What proportion of the population has an IQ below 95?
  • What is the IQ value at the 85 percentile?
  • The definition of genius is a difficult one. One suggestion is a score higher than 136 on an IQ test, while another is the top 0.1% of the distribution. Calculate the probability of a value of 136 or higher. Compare that that to the alternative definition of the top 0.1% (this is a probability of 0.001).

9.

In a sampling distribution for a mean, if we know the population mean and standard deviation (μ and σ), then the distribution of the sample means follows a normal distribution, with a mean equal to μ and the standard deviation equal to σ/SQRT(n). Note that σ/SQRT(n) is called the standard error. Based on this information, if we took a sample of size n (n is given in the problem as some number), what is the probability that the sample mean is greater than (or less than) some value? This is simply a z-score and normal distribution problem, similar to what we did earlier. However, there is one important change with these problems. Now the denominator of the z-score is the standard error and not σ. That said, answer the following questions.

A manufacturing process produces a product that contains an average of 3.5 liters of liquid with a standard deviation of 0.25 liters (i.e., μ = 3.5 and σ = 0.25). If the plant manager takes a sample of 64 observations, what is the probability that:

  • The sample mean is greater than 3.55
  • The sample mean is less than 3.6
  • The sample mean is between 3.45 and 3.5
  • The sample mean is exactly equal to 3.5

10.

In a sampling distribution for a mean, if we know the population mean and standard deviation (μ and σ), then the distribution of the sample means follows a normal distribution, with a mean equal to μ and the standard deviation equal to σ/SQRT(n). Note that σ/SQRT(n) is called the standard error. Based on this information, if we took a sample of size n (n is given in the problem as some number), what is the probability that the sample mean is greater than (or less than) some value? This is simply a z-score and normal distribution problem, similar to what we did earlier. However, there is one important change with these problems. Now the denominator of the z-score is the standard error and not σ. That said, answer the following questions.

Assume the systolic blood pressure of young adults in the U.S. aged 20 to 30 years follows a normal distribution, with μ = 113.7 and σ = 11.7. If we take a random sample of 150 young adults, what is the probability that:

  • The sample mean is between 113 and 115
  • The sample mean is less than 111
  • The sample mean is greater than 111
  • The sample mean is greater than 116.5

11.

Suppose we work for a company that makes a popcorn product containing 1.2 ounces of unpopped kernels in a microwavable bag. However, no manufacturing process is perfect, and there is variability from bag to bag, which the factory manager seeks to keep to a minimum. Bags that are under-filled can lead to consumer complaints and lawsuits, while bags that are overfilled can result in lost profits and affect the quality of the popping process. Based on previous experience, the distribution of the popcorn bags is distributed normally with a mean of 1.21 oz. and a standard deviation of 0.22, bag~N(1.21, 0.22).

  • Suppose the manger takes a sample of 16 bags and observes a sample mean of 1.20 with a standard deviation of 0.20. One of the bags in the sample weighs 1.3 ounces. Calculate a z-score for the value of 1.3 and interpret its meaning.
  • The manager asks the following question: “If the mean and standard deviation of the population are true as given (i.e., μ = 1.21 and σ = 0.22) and I took a random sample of 16 bags, what is the probability that the sample mean would be greater than 1.3 oz.?” Note: This problem is a sampling distribution problem. Use the standard error in calculating a z-score.
  • The manager asks the same question as in the previous problem, but in reference to 49 bags. He chooses a larger sample size. Compare the probabilities from using a sample of 16 bags and to 49 bags. Explain why the answers are different.

12.

The data below are means taken from random samples from a population that is distributed as a uniform continuous distribution with parameters 0 and 10. A uniform continuous distribution would generate a histogram that looks like a rectangle. With parameters 0 and 10 the mean of this distribution is 5 and the standard deviation is 2.89 (think of these as μ = 5 and σ = 2.89). A uniform distribution is decidedly not normal or bell-shaped and, as such, provides a good illustration of whether the sampling distribution resembles a normal distribution. Samples of 36 observations were randomly drawn, and the mean was calculated for each sample. This was done in Excel using the RAND function. To demonstrate sampling distributions, 36 different sample means were examined (below). This is a small sample of the sample means, but nonetheless, it gives us insight into sampling distribution theory.

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