PSYC 208 - Psychology Statistics

What are the real limits? 0.1

0.05 & 0.15

3 properties of standard scores

1. The mean of a set of z-scores is always zero 2. The SD of a set of standardized scores is always 1 3. The distribution of a set of standardized scores has the same shape as the unstandardized scores

Purpose of z-scores

1. identify and describe location of every score in the distribution. 2. Standardize an entire distribution -Sign tells whether score is located above or below the mean -Number tells distance between score and mean in standard deviation units

What are the real limits? 12.6

12.55 & 12.65

What are the real limits? 15.2

15.15 & 15.25

What are the real limits? 170.26

170.255 & 170.265

What are the real limits? 170.261

170.2605 & 170.2615

What are the real limits? 32

31.5 & 32.5

Notation: σ^2

Because the standard deviation is the square root of the variance, we write the variance of a population as σ^2

Learning check #1 T/F: The computational & definitional formulas for SS sometimes give different results.

False

Learning check #11 True/False: Σx^2=(Σx)^2

False

Learning check #2 T/F You can determine how many individuals had each score from a grouped frequency distribution

False

Learning check #2 T/F: If all the scores in a data set are the same, the Standard Deviation is equal to 1.00.

False

Learning check #3 True/False: When sample differs from the population there is a systematic difference between groups

False

Learning check #4 T/F: It is possible for more than 50% of the scores in a distribution to have values above the median

False

Learning check #6 T/F: The mean uses all the scores in the data, so it is the best measure of central tendency for skewed data.

False

Learning check #3: T/F:A score close to the mean has a z-score close to 1.00

False Scores close to 0 have z-scores close to 0.00

Learning Check #8 True/False Variables that cannot be measured directly are not real.

False- Constructs (internal states) can only be observed indirectly, but are the subject of much research

Learning check #5 True/False: All research methods have an independent variable

False- Correlational methods don't need an independent variable

Learning Check #6 True/False All research methods can show cause-and-effect relationships

False- Only experiments control the influence of participants and environmental variables

Learning check #8 T/F: A biased statistic has been influenced by researcher error

False: Bias refers to the systematic effect of using sample data to estimate a population parameter

Learning check #4 T/F: A treatment center for children measured the marital status of their parents (single, married, divorced, etc.) A histogram would be appropriate for these data.

False: Marital status is a nominal variable; a bar graph is needed

Learning check #6 T/F: The standard deviation is the distance from the Mean to the farthest point on the distribution curve

False: The standard deviation extends from the mean approximately halfway to the most extreme score

Standard deviation formula

Standard deviation= √ (SS/N)

Learning check #2 True/False: Most research studies use data from samples

TRUE

7. A population of scores has µ = 50 and σ = 12. If you subtract five points from every score in the population, then the new standard deviation will be _____. a. 7 b. insufficient information, cannot be determined c. 45 d. 12

a. 7 subtract 5 from σ

9. What is the shape of the distribution for the following set of data? Scores: 1, 2, 2, 2, 2, 3, 3, 4, 5, 6 a. positively skewed b. symmetrical c. rectangular d. negatively skewed

a. positively skewed

9. Which of the following is true for a symmetrical distribution? a. the mean, median, and mode are all equal b. mean = mode c. median = mode d. mean = median

a. the mean, median, and mode are all equal

2. A researcher is measuring problem-solving times for a sample of n = 20 children. However, one of the children fails to solve the problem so the researcher has an undetermined score. What is the best measure of central tendency for these data? a. the median b. the mode c. the mean d. Central tendency cannot be determined for these data.

a. the median

1. What is the value of ΣX+1 for the following scores? Scores 3,0,5,2 a) 14 b) 11 c) 32 D) 20

b) 11 (Take each of the #'s and add them together to =10 then +1=11)

3. Organizing a set of scores into a table or graph would be an example of using________. a) inferential statistic b) descriptive statistic c) population statistics d) sample statistics

b) descriptive statistic

Learning check #10 Σx^2+47 instructs you to .... a) square each score & add 47 up to it, then sum those #'s b) square each score, add up the squared scores, then add 47 to that sum c) add 47 to each score, square the result, and sum those #'s d) add up the scores, square that sum, and add 47 to it

b) square each score, add up the squared scores, then add 47 to that sum

7. What additional information is obtained by measuring on a interval scale compared to a ordinal scale? a) the direction of the differences b) the size of the differences c) whether the measurements are the same or different d) none of the above

b) the size of the differences

10. A skewed distribution typically has _____ distinct tail(s) and a normal distribution has ____ distinct tail(s). a. 1, 1 b. 1, 2 c. 2, 2 d. 2, 1

b. 1, 2

Learning check #1: A z-score of z = +1.00 indicates a position in a distribution ____ a. Above the mean by 1 point b. Above the mean by a distance equal to 1 standard deviation c. Below the mean by 1 point d. Below the mean by a distance equal to 1 standard deviation

b. Above the mean by a distance equal to 1 standard deviation

Learning check #2 A sample of n = 7 scores has M= 5. All of the scores are doubled.What is the new mean? a. M=5 b. M=10 c. M=25 D. More information is needed to compute M

b. M=10

5. A population has SS = 30 and σ^2 = 6. How many scores are in the population? a. cannot be determined without additional information b. N = 5 c. N = 6 d. N = 180

b. N = 5 σ^2=SS/N 6=30/N N=5

3. A population with a mean of μ = 6 has ΣX = 42. How many scores are in the population? Select one: a. N = 252 b. N = 7 c. cannot be determined from the information given d. N = 6/42 = 1/7

b. N = 7

4. For a population of N = 10 scores, you first measure the distance between each score and the mean, then square each distance and find the sum of the squared distances. What value have you calculated? a. none of the other choices is correct b. SS c. the population variance d. the population standard deviation

b. SS SS= Σ(x-µ )^2

5. The classrooms in the Psychology department are numbered from 100 to 120. A professor records the number of classes held in each room during the fall semester. If these values are presented in a frequency distribution graph, what kind of graph would be appropriate? a. a histogram or a polygon b. a bar graph c. a polygon d. a histogram

b. a bar graph

6. What kind of frequency distribution graph shows the frequencies as bars that are separated by spaces? a. a polygon b. a bar graph c. all of the above d. a histogram

b. a bar graph

Learning check #5 A distribution of scores shows Mean = 31 and Median = 43. This distribution is probably a. positively skewed b. negatively skewed c. bimodal d. open-ended

b. negatively skewed

7. In a positively skewed distribution, scores with the highest frequencies are _____. a. represented at two distinct peaks b. on the left side of the distribution c. in the middle of the distribution d. on the right side of the distribution

b. on the left side of the distribution

4. A population of N = 10 scores has a mean of µ = 80. If 5 points are added to every score in the distribution, what is the value of the new mean? a. still µ = 80 b. µ = 85 c. µ = 75 d. µ = 130

b. µ = 85

Learning Check #7 A study assesses the optimal size (# of other members) for study groups. The variable "Size of group" is a) discrete and interval b) continuous and ordinal c) discrete and ratio d) continuous and interval

c) discrete and ratio

8. After measuring 2 individuals, a researcher can say that Tom's score is 4 points higher that Bill's. The measurements must come from a(n) ________ scale. a) ordinal b) nominal c) interval or ratio d) interval

c) interval or ratio

2. A researcher is curious about the average monthly cell phone bill for high school students in the state of Florida. If the average could be obtained, it would be an example of a ______. a) statistic b) sample c) parameter d) population

c) parameter

Learning check #3 A Grouped Frequency Distribution table has categories 0-9, 10-19, 20-29, and 30-39. What is the width of the interval 20-29? a. 9 points b. 9.5 points c. 10 points d. 10.5 points

c. 10 points (29.5-19.5= 10)

Learning check #4: For a population with μ = 50 and σ = 10, what is the X value corresponding to z=0.4? a. 50.4 b. 10 c. 54 d. 10.4

c. 54 z= (x-µ)/σ SO x=µ=zσ x=50+ (0.4)(10) x=54

Learning check #4 A sample of four scores has SS = 24.What is the variance? a. 6 b. 7 c. 8 d. 12

c. 8

8. Which of the following is true for most distributions? a. Around 70% of the scores will be located within one standard deviation of the mean. b. Around 50% of the scores will be located within one standard deviation of the mean. c. Around 30% of the scores will be located within one standard deviation of the mean. d. Around 90% of the scores will be located within one standard deviation of the mean

c. Around 30% of the scores will be located within one standard deviation of the mean.

Learning check #3 The standard deviation measures.... a. sum of squared deviation scores b. standard distance of a score from the mean c. Average deviation of a score from the mean d. Average squared distance of a score from the mean

c. Average deviation of a score from the mean

5. A sample has a mean of M = 72. If one person with a score of X = 58 is removed from the sample, what effect will it have on the sample mean? a. cannot be determined from the information given b. The sample mean will remain the same. c. The sample mean will increase. d. The sample mean will decrease.

c. The sample mean will increase.

4. If the following distribution was shown in a histogram, the bar above the 15-19 interval would reach from _____ to _____. a. X = 15.5 to X = 19.5 b. X = 15.0 to X = 19.0 c. X = 14.5 to X = 19.5 d. X = 15.5 to X = 18.5

c. X = 14.5 to X = 19.5

6. A population of N = 10 scores has a mean of μ = 6. After one score is removed, the mean is found to be M = 5. What is the value of the score that was removed? Select one: a. X = 10 b. X = 5 c. X = 15 d. X = 3

c. X = 15

6. If sample variance is computed by dividing SS by df = n - 1, then the average value of the sample variances from all the possible random samples will be _______ the population variance. a. larger than b. exactly equal to c. smaller than d. unrelated to

c. smaller than

8. In a normal shaped distribution, ______. a. the scores pile up on the left-hand side and taper off to the right. b. the scores pile up on the right-hand side and taper off to the left. c. the scores pile up in the middle and taper off symmetrically to both sides. d. the scores are evenly distributed across the entire scale of measurement.

c. the scores pile up in the middle and taper off symmetrically to both sides.

Learning check #7 A population has a μ = 6 and σ = 2. What is the shape of the resulting distribution? a. μ = 60 and σ = 2 b. μ = 6 and σ = 20 c. μ = 60 and σ = 20 d. μ = 6 and σ = 5

c. μ = 60 and σ = 20

A set of scores ranges from a high of X =48 to a low of X = 7. If these scores are placed in a grouped frequency distribution table with an interval width of 5 points, the bottom interval in the table would be _______. a) 7-11 b) 5-10 c) 7-12 d) 5- 9

d) 5- 9

Learning Check #1 A researcher is interested in the effect of amount of sleep on high school students' exam scores. A group of 75 high school boys agree to participate in the study. The boys are ..... a) a statistic b) a variable c) a parameter d) a sample

d) a sample

5. Real limits are important whenever you are measuring a(n) ________ variable. a) discrete b) dependent c) independent d) continuous

d) continuous

Learning check #4 Researchers observed the students exam scores were higher the more sleep they had the night before. This study is... a) descriptive b) experimental comparison of groups c) non-experimental group comparison d) correlational

d) correlational

6. An operational definition defines a hypothetical construct ______. a) abstractly, like a construct b) all of the above c) conceptually, like a dictionary definition d) in terms of methods used to measure a manipulate it

d) in terms of methods used to measure a manipulate it

4. The average verbal SAT score for the entire class of entering freshmen is 530. However, if you select a sample of 20 freshman and compute their average verbal SAT score you probably will not get exactly 530. What statistical concept is used to explain the natural difference that exists between a sample mean and the corresponding population mean. a) statistical error b) parametric error c) inferential error d) sampling error

d) sampling error

1. For a population with µ = 40 and σ = 8, what is the z‑score corresponding to X = 46? a. +1.00 b. +1.50 c. +0.50 d. +0.75

d. +0.75 z= (x-µ)/σ z= (46-40)/8 z=6/8 z=0.75

3. What is the value of SS (sum of squared deviations) for the following sample? Sample: 2, 3, 4, 7 a. 78 Incorrect b. 14/3 = 2.67 c. 72 d. 14

d. 14

8. What is the median for the following set of scores? Scores: 1, 2, 6, 11, 17 a. 8.5 b. 8 c. 4 d. 6

d. 6

1. What is the mean for the following scores? Scores: 1, 6, 14 a. 10.5 b. 3 c. 6 d. 7

d. 7

Learning check #1 A sample of n = 12 scores has a mean of M = 8. What is the value of ΣX for this sample? a. 1.5 b. 4 c. 20 d. 96

d. 96

7. In a population of N = 6, five of the individuals all have scores that are exactly 1 point above the mean. From this information, what can you determine about the score for the sixth individual? Select one: a. It is also above the mean by 1 point. b. There is not enough information to describe the 6th score. c. It is below the mean by 1 point. d. It is below the mean by 5 points.

d. It is below the mean by 5 points.

1. For any set of data, the sum of the deviation scores will always be _____. a. less than zero b. impossible to determine without more information c. greater than zero d. equal to zero

d. equal to zero

10. For a negatively skewed distribution with a mode of X = 25 and a median of 20, the mean is probably _____. a. cannot be determined from the information given b. greater than 25 c. between 20 and 25 d. less than 20

d. less than 20

9. The smallest score in a population is X = 5 and the largest score is X = 10. Based on this information, you can conclude that ______. a. None of the other choices is correct. b. the population mean is between 5 and 10, and the standard deviation is less than 6. c. the population standard deviation is smaller than 6. d. the population mean is somewhere between 5 and 10.

d. the population mean is somewhere between 5 and 10.

10. For a particular sample, the largest distance (deviation) between a score and the mean is 11 points. The smallest distance between a score and the mean is 4 points. Therefore, the standard deviation _____. a. It is impossible to say anything about the standard deviation. b. will be greater than 11 c. will be between 4 and 11 d. will be less than 4

d. will be less than 4

df's formula

degrees of freedom df= n-1

Statistic notation N

is the # of scores in a population

Statistic notations n

is the # of scores in a sample

Formula for standard deviation of sample

s=√(SS/(n-1))

formula for variance of sample

s^2= SS/ (n-1)

__ X

same as M; mean of a sample

Notation: s

standard deviation of sample

Statistic notations Σ

summation sign

µ

the mean of a population

M

the mean of a sample

Notation: σ

the standard deviations of a population

Notation: s^2

variance of a sample

formula to find z

z= (x-µ)/σ SO x=µ=zσ (x-µ)= deviation score σ = expresses deviation in standard deviation units

2. Which set of scores has the smallest standard deviation? a. 27, 105, 10, 80 b. 145, 143, 145, 147 c. 5, 11, 42, 22 d. 11, 17, 31, 53

a. 27, 105, 10, 80

1. A sample of n=20 scores ranges from a high of X=9 to a low of X=3. If these scores are placed in a frequency distribution table, how many X values will be listed in the first column? a) 7 b) 9 c) 20 d) 6

a) 7

2. For the following distribution of quiz scores, how many individuals took the quiz? X f 5 2 4 4 3 2 2 1 a)n=9 b) cannot be determined c)n=5 d)n=15

a)n=9

Learning check #12 True/False: (Σx)(Σx)=(Σx)^2

True

Learning check #2: T/F: A negative z-score always indicates a location below the mean

True

Learning check #3 T/F: It is possible for more than 50% of the scores in a distribution to have values above the mean.

True

Learning check #7 T/F: The mean and median have the same values, so the distribution is probably symmetrical

True

Learning Check #9 True/False: Research measurements are made using specific procedures defined ahead of time.

True- operational definitions assure consistent measurements and serve as definitions of constructs

Learning check #9 T/F: On average, an unbiased sample statistic has the same value as a population parameter

True: Each sample's statistic differs from the population parameter, but the average of all samples will equal the parameter

Learning check #5 T/F A sample systematically has less variability than a population

True: Extreme scores affect variability, but are less likely to be included in a sample

Learning check #5 T/F: A treatment center for children measured the time they spent playing with other children (in minutes). A histogram would be appropriate for these data.

True: is measured continuously and is an interval variable

Learning check #1 T/F You can determine how many individuals had each score from a frequency distribution table

Ture

Variance formula

Variance= SS/N