Statistics and Probability Answers

1. Information from the Department of Motor Vehicles indicates that the average age of  licensed drivers is 38.6 years with a standard deviation of 10.4 years. Assume that the  distribution of the driver’s ages is normal.  

a. What proportion of licensed drivers are from 25 to 45 years old?

b. Determine the ages of licensed drivers separating the upper 10% and lower 10%  percent of the population.

2. Weights of newborn babies in a particular city are normally distributed with a mean of  3380 g and a standard deviation of 475 g. 

a. A newborn weighing less than 2100 g is considered to be at risk, because the  mortality rate for this group is very low. If a hospital in the city has 500 births in a  year, how many of those babies are in the “at-risk” category?

b. If we redefine a baby to be at risk if his or her birth weight is in the lowest 3%,  find the weight that becomes the cutoff separating at-risk babies from those who  are not at risk.

c. If 20 newborn babies are randomly selected as a sample in a study, find the  probability that their mean weight is between 3200 g and 3500 g.

3. Over the past 30 games between team A and team B, team A has won 14 times, team B has won 11 times and the game has ended in a draw 5 times. If these two teams played 9  games this season, what is the probability that team �� would win 5 games, team �� would  win 3 games, and the remaining game would be a draw?

4. The formula for the standard error (standard deviation of the distribution of sample  means) implies that as the sample size (n) increases, the size of the standard error  decreases. Explain the role of the standard error in comparing the sample mean and the  population mean. Use the definitions and concepts on sample means distribution and  standard error and show examples comparing the sample mean and the population mean  of a distribution when the standard error changes in value to express your answer.

Use an analogy to explain a Type I error and a Type II error and discuss its significance in hypothesis testing.

The percentage of physicians who are women is 27.9%. In a survey of physicians  employed by a large university health system, 60 of 142 randomly selected physicians  were women. Is there sufficient evidence at the 0.05 level of significance to conclude that  the proportion of women physicians at the university health system exceeds 27.9%?

a. State the hypothesis and identify the claim of the researcher.

b. Find the critical value(s). 

c. Compute the test value. 

d. Make a decision on the null hypothesis. 

e. Make a decision on the claim of the researcher.

Unit 5 deals with two types of discrete random variables, the Binomial and the Poisson, and two types of continuous random variables, the Uniform and the Exponential. Depending on the context, these types of random variables may serve as theoretical models of the uncertainty associated with the outcome of a measurement.

Give an example, USING YOUR OWN WORDS (NOT TEXT COPIED FROM THE INTERNET), of how either the Poisson or the Exponential distribution could be used to model something in real life (only one example is necessary). You can give an example in an area that interests you (a list of ideas is below). Give a very rough description of the sample space.

If you use an idea from another source, please provide a citation in the sentence and a reference entry at the end of your post. Include a citation even if you paraphrase from a website. Please do not copy blocks of text from the Internet--try to use your own words.

When forming your answer to this question you may give an example of a situation from your own field of interest for which a random variable, possibly from one of the types that are presented in this unit, can serve as a model. Discuss the importance (or lack thereof) of having a theoretical model for the situation. People can use models to predict business conditions, network traffic levels, sales, number of customers per day, rainfall, temperature, crime rates, or other such things.

A particular type of smart phone comes in a regular-size version and a “plus-size” version. 60% of all customers prefer the “plus-size” version.

(a) Among 10 randomly selected customers who want this type of smart phone, what is the probability that at least eight want the “plus-size” version?

(b) Among 10 randomly selected customers, what is the probability that the number of customers who want the “plus-size” version is within 1 standard deviation of the mean value? [Hint: what probability distribution should be used, and what are the associated mean and standard deviation?]

(c) The store currently has 7 smart phones of each version. What is the probability that all of the next 10 customers who want this smart phone can get the version they prefer from the current stock? [Hint: what is the value range for # of customers who want the “plus-size” given the restricted stock?]

(d) The store is retailing the regular-size smart phone for $400 and the “plus-size” one for $600, and the store has 20 smart phones for each version in stock. What is the expected revenue from the purchases of the next 3 customers?

Rachel put 3 red marbles, 2 blue marbles, 1 yellow marble, and 4 green marbles into a bag. All the marbles were the same shape and size. Without looking, Rachel pulled 2 marbles out of the bag without putting the first back in. Which of the following would be an impossible outcome for this event? 

The weights of a normally distributed group of adults participating in a fitness test has an average of 60kg and a standard deviation of 8kg.

a. What is the probability that an adult’s weight is in between 55-65kg?

b. What raw score divides the group such that 55% is above it?

What is the probability that an adult’s weight is in between 55-65kg?

The weights of 1,000 children, in average, is 51kg with standard deviation of 11kg. Suppose the weights are normally distributed, how many children weigh between 31kg and 69kg

The Dean the School of education in a University wants to determine whether there are statistically significant difference of opinion among the different cadres of academic staff members at the university concerning a proposed curriculum change in which postgraduate students are to be taught research methods on-line. Interviewing a sample of 313 members of the academic staff constituting 104 Lecturers, 131 Senior Lecturers, and 78 Professors, the Dean obtained the results shown in the following table:

Rank

Response

Lecturer

Senior Lecturer

Professor

Total

Against

47

34

14

95

Not committed

41

49

29

119

In support

16

48

35

99

Total

104

131

78

313

Compute a chi-square test for the above data and draw conclusion at α = .05                              

4) The mean I.Q of a sample of 1600 children was 99. It is likely that this was a random sample from a population with mean I.Q 100 and standard deviation 15.

 Required:

Using appropriate statistics, deduce whether there is any statistically significant difference between the two means if z critical value at α = .05 is 1.96