Statistics

When is the mean an appropriate measure of central tendency?

The mean is appropriate for interval or ratio data that is continuous, symmetrical and lacks significant outliers

When is the median an appropriate measure of central tendency?

The median is appropriate for continuous data that might be skewed (asymmetrical), based on ranks, or contain extreme values

When is the mode an appropriate measure of central tendency?

The mode is appropriate for categorical variables or for continuous data sets where one value dominates the others

What is operationalization?

The process of specifying how a concept will be defined and measured

You are taking a multiple-choice test with 10 items, in which there is no penalty for wrong answers. There are 5 possible answers for each question, so just by guessing, you have a 20% chance of getting the answer right on each question. Assuming you simply guess at the right answers, what is the probability that you will get 3 or more right?

Use binomial distribution. n = 10, k = 3, p = 0.2. P(k = 0) = 0.11. P(k = 1) = 0.27 P(k = 2) = 0.30 P(k >= 3) = 1 - P(k <= 2) = 1 - (0.11 + 0.27 + 0.30) = 0.32

You are taking a multiple-choice test with 10 items, in which there is no penalty for wrong answers. There are 5 possible answers for each question, so just by guessing, you have a 20% chance of getting the answer right on each question. Assuming you simply guess at the right answers, what is the probability that you will get exactly 3 right?

Use binomial distribution. n = 10, k = 3, p = 0.2. P(k = 3;10,0.20) = (10 3)0.2^3(1-0.2)^7 = 0.20 = 20%

List the potential bias to be aware of in this scenario: A program intended to improve scholastic achievement in high school students reports success b/c the 40 students who completed the year-long program (of 100 who began it) all showed significant improvement in their grades and scores on standardized tests of achievement.

Informative censoring: Those who completed might have been more intelligent, or those that dropped out might not have benefited from the program

What are the variance and standard deviation of the following data set for both population and sample data: [1,3,5]?

Mean = 3, pop variance = (1-3)^2+(3-3)^2+(5-3)^2/3 = 8/3 = 2.67, pop standard deviation = 1.63, sample variance = 8/(3-1) = 4, sample standard deviation = 2

What are the 4 common data types used by statisticians?

Nominal, ordinal, interval and ratio

If order is not significant, how many ways are there to select a subset of 5 students from a class of 20?

20C5 = 20!/5!(20 - 5)! = 20P5/5! = 1860480/120 = 15504

Should a data set of the majors elected by 10,000 entering freshmen at a university be displayed as a bar chart or a histogram? Why?

Bar chart, because the information is categorical and has a restricted set of possible values

Why is the central limit theorem of primary importance to the practice of inferential statistics?

The central limit theorem states that the sampling distribution of the sample mean approximates the normal distribution, regardless of the distribution of the population from which the samples are drawn if the sample size is sufficiently large. This is important, b/c if the sample size is large enough we can make inferences about the population

What type of sampling is described in this scenario: The goal is to collect information on iron deficiency, obtained through blood tests, on the US population. A sampling plan is devised so that units are selected from successively smaller regions of the country. Regions are selected at random, then states w/in regions and so on down to individual houses w/in census block groups?

Cluster sampling

Should a data set of the heights (in cm) of 10,000 entering freshmen at a university be displayed as a bar chart or a histogram? Why?

Histogram, because these measurements are continuous and have a large number of possible values

What is the probability of drawing two black cards in a row from a regular 52 card deck? Write the formula first (intersection of non-independent events)

P(E∩F) = P(E) x P(F|E) = 26/52 X 25/51 = 0.5 X 0.49 = 0.245

80 students are attending a conference. 40 male and 40 female. 30 boys are math majors. 20 girls are math majors. If you pick a math major at random, what is the probability that they are male? (Bayes Theorem)

P(male) = 0.5, P(~male) = 0.5, P(math|male) = 0.75, P(math|~male) = 0.5. P(male|math) = 0.75*0.5/0.75*0.5 + 0.5*0.5 = 0.6 = 60%

What type of sampling is described in this scenario: The goal is to find out how elementary school students are reacting to a recently appointed principal. The researcher wants to include equal numbers of male and female students in the sample, so the interviewer is sent to the school with instructions to interview 10 male students and 10 female students from among those on the playground after school one day?

Quota sampling (and convenience sampling)

What is the difference btw interval and ratio data?

Ratio data has a natural zero point

If I flip a fair coin twice, what is the probability that I will get at least one head? List the sample size, event and solution

S = {(h, h), (h, t), (t, h), (t, t)}, E = {(h, h), (h, t), (t, h)}, P(E) = 3/4 = 0.75 = 75%

If I roll a dice once, what is the probability of getting a number below 5? List the sample size, event and solution

S = {1 - 6}, E = (E = 1)U(E = 2)U(E = 3)U(E = 4), P(E) = 4/6 = 0.67 = 67%

If I draw a black card from a deck, what is the probability that the suit is clubs? List the sample size, event and solution

S = {26 cards}, E = (clubs|black card), P(E) = 13/26 = 0.5 = 50%

If I draw one card from a deck, what is the probability that it will be a face card or a black card? List the sample size, event and solution

S = {52 cards}, E = (E = black card (26))U(E = k, q, or j (12)), P(E) = (26/52) + (12/52) - ((26/52)*(12/52)) = 0.616

If I draw one card from a deck, what is the probability that it will be a black face card? List the sample size, event and solution

S = {52 cards}, E = (E = black card (26))n(E = k, q or j (12)), P(E) = (26/52)*(12/52) = 0.115

If I draw one card from a deck, what is the probability that it will be red? List the sample size, event and solution

S = {52 cards}, E = {red card}, P(E) = 26/52 = 0.5 = 50%

What type of sampling is described in this scenario: The goal is to learn about the domestic life of police officers working in a major city, including how home life is affected when the police officer's spouse is employed outside the home. A complete list of all men and women working as police officers in the city is available, and a computer draws a random sample of 200 from the list. Members of the sample are then interviewed by telephone?

Simple random sampling

List the potential bias to be aware of in this scenario: A university reports the average annual salary of its graduates as $120,000, based on responses to a survey of contributors to the alumni fund

Social desirability bias, selection bias and nonresponse bias: The reported average annual salary is probably an overestimate of the true value b/c subscribers to the alumni mag were probably among the more successful grads and people who felt embarrassed about their low salary were less likely to respond

List the potential bias to be aware of in this scenario: A manager is concerned about the health of his employees, so he institutes a series of lunchtime lectures on topics such as health eating, the importance of exercise, and the deleterious health effects of smoking and drinking. He conducts an anonymous survey (using a pencil-and-paper questionnaire) of employees before and after the lecture series and finds that the series has been effective in increasing healthy behaviors and decreasing unhealthy behaviors

Social desirability bias: The employees know the employer cares about the subject

What type of sampling is described in this scenario: A factory supervisor is concerned that the quality of parts produced might not be equal on all shifts or at all times w/in a shift. (Factory operates 24 hours per day.) A sampling plan is devised to collect samples of 30 parts at 9 times during the work day, with the times of collection selected randomly w/in time blocks for each of the three daily shifts. W/in each shift, one sample will be drawn w/in the first 2 hours, one w/in the middle 6 hours and one w/in the last 2 hours?

Stratified sampling

Calculate the Z-score of the value 108, assume it came from a normal population with mean = 100, variate = 2

Z = (108 - 100)/2 = 4.00

Compute the Z-statistic for the following sample (mean = 42, n = 35), which was drawn from a population with a mean of 40 and a standard deviation of 5

Z = (42 - 40)/(5/square root of 35) = 2.37

You are a school principle. One of your students is given an IQ test and receives a score of 80. You know that for the students age group in the population at large, scores on this test are distributed normally (mean = 100, variant = 15). What statistics will help you interpret the students score?

Z-score & standard normal distribution table (Figure D-3 n Appendix D). Z = (80 - 100)/15 = -1.33. P(Z <= -1.33) = 1 - P(Z >= -1.33) = 1 - (0.50000 + 0.40824) = 0.09176

What is the mean and median of this data set: [1,7,21,3,-17]?

n = 5, ordered set = [-17,1,3,7,21], mean = (-17+1+3+7+21)/5 = 3, median = (5 + 1)/2 = 3rd = 3

What is the median of this data set: [7,15,2,6,12,0]?

n = 6, ordered set = [0,2,6,7,12,15], 6/2 = 3, 3 + 1 = 4, median = (6+7)/2 = 6.5

What is the median of this data set: [8,3,2,7,6,9,1,2,1]?

n = 9, ordered set = [1,1,2,2,3,6,7,8,9], (9 + 1)/2 = 5, median = 3