Psych 248 Final Exam Practice 1.
Standard deviation
- Distance from the mean (X-µ) EXAMPLE: If you have µ=50 and score of X=53 Your deviation is X-µ or 53-50=3 If your score is X=45 Your deviation is X-µ or 45-50=-5 -The most commonly used measure of variability -Measures the standard distance between all scores and the mean -Determines if scores are near or far from the mean -Quantitative data -Always positive
The Mean
- The mean is the most commonly used measure of central tendency -The mean is obtained by computing the sum, or total, for the entire set of scores, then dividing this sum by the number of scores. Mean=ΣX/N Sample 3, 7, 4, 6 The mean is X= ∑X/n= 20/4=5 μ= Population mean X or M = Sample mean
What happens when you are changing the mean with a constant.
-Add a score above the mean > the mean will increase -Add a score below the mean > the mean will decrease -Delete a score below the mean > the mean will increase -Delete a score above the mean > the mean will decrease -Add/delete a score equal to the mean > Mean does not change
3 ways of describing the distribution of data
-Central tendency-measures where the center of the distribution -Variability - if scores are widely spread or clustered together -Shape- the relationship between X and Y values on the graph
3 ways of describing the way the distribution of data
-Central tendency-measures where the center of the distribution -Variability - if scores are widely spread or clustered together -Shape- the relationship between X and Y values on the graph -All distributions are either symmetrical or skewed
What is contained in Central Tendency
-Mean -Weighted mean -Median -Mode
Variability
-Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together - A measure of variability provides an objective description of the differences between the scores in a distribution by measuring the degree to which the scores are spread out.
The Median
-The median is defined as the midpoint of the list -Take the mean of the two middle scores 4+5/2=4.5 -The median tends to stay in the "center" of the distribution
The Mode
-The most frequently occurring category or score in the distribution -The most frequently occurring category or score in the distribution -Unimodal distributions (One mode for either a normal or skewed distribution) -Bimodal distributions -Multimodal distributions
Frequency distribution graphs
-Useful because they show the entire set of scores -Determination of the highest score, and the lowest score -Where the scores are centered -Shows whether the scores are clustered together or scattered over a wide range
Standard Deviation: Variance
-Variance =σ2 = SS/N σ2 = 40/5=8 -We need to take the square root of the variance √8=2.83=σ
The Interquartile Range
-When trying to avoid influence of extreme scores -The interquartile range -The distance covered by the middle 50% of the distribution (the difference between Q1 and Q3). IQR=Q3-Q1 -Locate boundary that -separates lowest and highest 25% of the distribution -Because it only represents the middle 50% it doesn't represent entire data set -But better for data with outliers semi- interquartile range = Q3-Q1 / 2
. Thirty AA batteries were tested to determine how long they would last. The results, to the nearest minute, were recorded as follows: 423, 369, 387, 411, 393, 394, 371, 377, 389, 409, 392, 408, 431, 401, 363, 391, 405, 382, 400, 381, 399, 415, 428, 422, 396, 372, 410, 419, 386, 390 a. Create a simple frequency table distribution for these grouped data with five intervals
. Which interval had the largest frequency?
steps of hypotheses testing
1. state the hypotheses 2. set criteris for decision 3. compute test stats 4. make decision
Finding the middle 80% of SAT scores?
80% (0.8000) in the middle of the distribution can be split in half with 0.4000 on the left and 0.4000 on the right of the mean 1. Again choose closet number from table=0.3997 2. Corresponding z-score=1.28 +1.28 and -1.28 3. Back to X 1.28σ=1.28(100)=128 points 4. The right boundary is 500+128=628 5. The left boundary is 500-128= 372 6. Therefore the middle 80% of scores = 372 to 628 points
What is the variance for the following population scores? Scores: 5, 2, 5, 4 a) 1.5 b) 2 c) 6 d) 1.22
A 1.5
Calculate the variance for the following data Sample X M x - M (xi - M)^2 x1 17 x2 15 x3 23 x4 7 x5 9 x6 13 ∑
Answer: First, add your data points together: 17 + 15 + 23 + 7 + 9 + 13 = 84 Next, divide your answer by the number of data points, in this case six: 84 ÷ 6 = 14. Subtract the mean from each data point. - x 17 - 14 = 3 - x 15 - 14 = 1 - x 23 - 14 = 9 - x 7 - 14 = -7 - x - 14 = -5 - x 13 - 14 = -1 It's easy to check your work, as your answers should add up to zero. Square each result. Add these up to find the sum of all square values. 166 Use the formula s2 = 166/6-1 =33.2 Find the standard deviation of the above problem. S √33.2 5.76
An experiment was done with 30 participants to test the effect of the type of attachment style that one has (i.e., secure, anxious, or avoidant) on how long one (in years) is married to her/his spouse. Higher scores below indicate a greater number of years. Use an alpha level of .05 (p = .05). Secure Anxious Avoidant M1 = 10 M2 = 9 M3= 5 k = n1 = 10 n2 = 10 n3 = 10 N = T1 = 100 T2 = 90 T3 = 50 G = SS1=60 SS2= 110 SS3= 70 ∑X2 2300 a. What statistical test should you use to evaluate the hypothesis above? i. Simple linear regression ii. Pearson Product Moment Correlation iii. One-way independent ANOVA iv. T-test b. State the Hypothesis in words and notation Null Hypothesis (notation): ___________________________________ Null Hypothesis (words): _____________________________________ Alternative Hypothesis (notation): _____________________________ Alternative Hypothesis (words): _______________________________ c. Find: k ________, N ______, G ______ d. Find the critical value _____________ e. Compute the r of F statistic using the summary table below SS df MS Between treatments Within treatments Total f. Do you reject or retain the Null Hypothesis? Circle one: Reject Retain g. State your conclusion in terms of the study, use the proper statistics, and write in APA style. _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ ____________________________________________________
Answer: a - iii, b. µ1 = µ2 = µ3, All the means are the same, no notation for alternative, At least two of the means are different. C. k=3, N= 30, G 240. D. critical value is 3.35 e. SS df MS F Between treatments 140 2 70 7.87 Within treatments 240 27 8.9 Total 380 29 f. Reject g. A one-way ANOVA found a significant difference on the effect of attachment style and the number of years one has been married F(2, 27) = 7.87, p < .05.
A score 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? a) X = 3 b) X = 5 c) X= 15 d) X = 10
C x=15
Range
Distance between the largest number and the smallest number Gives us a very basic view of the data set EX: 3, 7, 12, 8, 5, 10 Xmax=12 Xmin = 3 Range= 12 - 3=9
Weighted Mean
Group #1 (n=12, M=6) Therefore the total ∑X=72 Group #2 (n=6, M=7) Therefore the total ∑X=42 M(w)= ∑(Mxn)/∑n Weighted mean = ∑X1 + ∑X2 / n1 + n2
State the number of participants when, dfBG = 4 dfE = 20 a. 2 b. 4 c. 6 d. 8
I apologize there was a mistake for this question, the answer is 25.
Descriptive statistics
Methods for organizing and summarizing data Mean, median, mode, skew, range etc...
Inferential statistics
Methods for using sample data to make general conclusions (inferences) about populations. -sample data provide only limited information about the population - sample statistics are generally imperfect representatives of the corresponding population parameters
The mean is appropriate for use with which of the following scales of measurement? a. Ordinal b. Nominal c. Interval d. Ratio
Ordinal - measurements where values convey order or rank alone Nominal - measurements in which number is assigned to represent something or someone. Interval - measurements that have no true zero and are distributed in equal units Ratio - measurements that have a true zero and are distributed in equal units
reject the null when
P value is less than alpha value ex: <.05
accept the null
P value is more than the alpha value ex: >.05
What are the many ways to represent variablitiy
Range interquartile range semi-interquartile range Variance Standard deviation
What are the steps used to construct a frequency table?
Step 1: Find the range - the difference between last number and first number Step 2: Find the intervals - The number in range divided by the intervals looking for Step 3: Draw the frequency table - use the above information.
Raw Scores
X = μ + zσ
A population od scores has μ = 44. In this population, a score of X = 40 corresponds to z = 0.50. What is the population standard deviation? a) 2 b) 4 c) 8 d) -8
a 2
Central Tendency
a statistical measure that determines a single value that accurately describes the center of the distribution and represents the entire distribution of scores.
19. What is indicated by a positive value fir a correlation? a) Increases in X tend to be accompanied by increases in Y. b) Increases in X tend to be accompanied by decreases in Y. c) A much stronger relationship exists than if the correlation were negative. d) A much weaker relationship exists than if the correlation were negative.
a) Increases in X tend to be accompanied by increases in Y.
Which of the following accurately describes the chi-square test for goodness of fit? a) It is similar to a correlation because it uses one sample to test a hypothesis about one population. b) It is similar to a correlation because it uses one sample to evaluate the relationship between two variables. c) It is similar to an independent-measures t test because it uses separate samples to evaluate the difference between separate populations. d) It is similar to both correlation and an independent-measures t test because it can be used to evaluate a relationship between variables or a difference between populations.
a) It is similar to a correlation because it uses one sample to test a hypothesis about one population.
For independent-measures ANOVA, individual differences contribute to the variance in the numerator and in the denominator of the F-ratio. What happens to the individual differences in the numerator if the F-ratio for repeated-measures ANOVA? a) They do not exist because the same individual participates in all of the treatments. b) They are measured and contribute to the variance in the denominator. c) They contribute to the variance in the numerator. d) They are amplified in the numerator of the ratio.
a) They do not exist because the same individual participates in all of the treatments.
3. In a correlational study, ________________________. a) a quantitative method of research in which you have 2 or more quantitative variables from the same group of participants, and you are trying to determine if there is a relationship between the 2 variables. b) a quantitative method of research in which you have 2 or more quantitative variables from the same group of participants, and you are trying to determine if the one variable causes the other variable. c) a qualitative method of research in which you have 2 or more qualitative variables from the same group of participants, and you are trying to determine if the one variable causes the other variable. d) a qualitative method of research in which you have 2 or more qualitative variables from the same group of participants, and you are trying to determine if there is a relationship between the 2 variables.
a) a quantitative method of research in which you have 2 or more quantitative variables from the same group of participants, and you are trying to determine if there is a relationship between the 2 variables.
. In ANOVA, 2 and 2 a. Estimate the size of the difference between groups b. Can be used to partition the total variance in scores into its components c. Are central elements of the structural model d. Identify which groups in the design are significantly different
a. Estimate the size of the difference between groups
6. What does decreasing the confidence interval do to precision and certainty? a. Increase precision, decrease certainty b. Increase precision and certainty c. Decrease precision, increase certainty d. Decrease precision and certainty
a. Increase precision, decrease certainty
An electronics firm gives a $1,500 bonus to every salaried employee at the end of the year. Compared to the original distribution of salaries (without the bonus), the new distribution has: a. a new mean and the same standard deviation b. a new mean and a new standard deviation c. the same mean and the same standard deviation d. the same mean and a new standard deviation
a. a new mean and the same standard deviation
14. For a two-way ANOVA, rejection of the overall hypothesis indicates that a. at least one pair of means differs from the other pairs b. at least several population means differ from each other c. at least some population means are equal d. all population means are different
a. at least one pair of means differs from the other pairs
Non-experimental
are similar to experiments because they also compare groups of scores, but it lacks true manipulation-should not use cause and effect statements Ex: testing drug on female and male brain
If a researcher is using a t-statistic to test a null hypothesis about a population, what information is needed from the population to calculate the t-statistic? a) You must know the population mean and the variance or standard deviation. b) The t-statistic does not require any information about the population. c) You must know the population mean. d) You must know the population mean and the variance.
b) The t-statistic does not require any information about the population
17. Which of the following describes the Pearson Correlation for the data? X 2 5 1 4 Y 5 3 4 2 a) It is positive b) It is negative c) It is close to zero d) It is equal to zero
b) negative
A set of X and Y scores has SSX = 20, SSY = 40. What is the slope for the regression equation? a) 4.0 b) 2.0 c) 0.25 d) 0.50
b)2.0
Appearance-related self-esteem (r = .40, p < .001) and media influence (r = .62, p < .001) and overall self-esteem (r = .08, p > .05) were tested to see if they correlated with consumer masculinity. Which variable was significant and had the strongest correlation? a. Appearance-related self-esteem b. Media influence c. Overall self-esteem d. Consumer masculinity
b. Media influence
The main advantage over an ANOVA over a t-test is that an ANOVA: a. Uses a higher p-value b. Reduces error variance when there are more than 2 levels in the independent variable by only conducting one test c. Can look at a numeric independent variable d. Can look at causation rather than correlation
b. Reduces error variance when there are more than 2 levels in the independent variable by only conducting one test
If a researcher rejects a null hypothesis, the researcher either a. is incorrect, or made a Type I error. b. is correct, or made a Type I error. c. is incorrect, or made a Type II error. d. is correct, or made a Type II error.
b. is correct, or made a Type I error.
A set of X and Y scores has MX = 3, SSX= 12, MY = 8, SSY = 56, and SP = 36. What is the regression equation for predicting Y from X? a. Ŷ = 12X - 36 b. Ŷ = 3X - 1 c. Ŷ = 4X + 3 d. Ŷ = 2x - 3
b. Ŷ = 3X - 1 Answer b - My - bmx 8 - 3 (3) = -1 sp/ssX = 36/12 = 3
8. What proportion of a normal distribution us located between z = +1.50 and z = -1.50? a) 0.9332 b) 0.4332 c) 0.8664 d) 0.0668
c 0.8664
For a population with μ = 80 and σ = 20, the distribution of sample means based on n = 16 will have an expected value of ___ and a standard error of ___. a) 5; 80 b) 20;20 c) 80; 5 d) 80; 1.25
c 80;5
10. A sample is selected from a population with μ = 46 and a treatment is administered to the sample. After treatment, the sample mean is M = 48 with a sample variance of s2 = 16. Based on this information, the size of the treatment effect, as measured by Cohen's d, is ___. a) d = 0.001 b) d = 0.125 c) d = 0.50 d) d = 0.25
c d=0.25
For a two-factor study with two levels of Factor A, two levels of Factor B, and a separate sample of n = 10 participants in each treatment condition. What is df WITHIN treatments? a) 9 b) 18 c) 36 d) 40
c) 36
9. Which of the following accurately describes the effect of increasing the sample size? a) Increases the standard error and has no effect on the risk of a Type 1 error. b) Increases the risk of a Type 1 error and has no effect on the standard error. c) Decreases the risk of standard error and has no effect on the risk of a Type 1 error. d) Decreases the risk of Type 1 error and has no effect on the standard error.
c) Decreases the risk of standard error and has no effect on the risk of a Type 1 error.
A sample of n = 20 scores has a mean of M = 45 and a standard deviation of S = 8. In this sample, what is the z-score corresponding to X = 57? a) Z = 0.50 b) Z = 1.00 c) Z = 1.50 d) Z = 2.00
c) Z = 1.50
A linear (simple or multiple) regression uses: a. Variance to calculate the statistic b. An F-ration as the statistic c. Looks at the effect of a numeric IV on a numeric DV d. Can evaluate the effect of more than one IV
c. Looks at the effect of a numeric IV on a numeric DV
What descriptive statistics do you use to describe the spread of a distribution? a. Mean b. Median c. Standard deviation d. A and B
c. Standard deviation
For an independent samples or (independent measures) research study, what is measured by Cohen's d (r2 )? a. The risk of a Type I error b. The risk of a Type II error c. The size of the difference between the two treatments d. Whether the difference between the two treatments is likely to have occurred by chance
c. The size of the difference between the two treatments
A two-way ANOVA should be conducted instead of a one-way ANOVA when: a. The independent variable has more than 2 levels b. There is more than one dependent variable c. There is more than one independent variable d. There is a categorical dependent variable
c. There is more than one independent variable
A psychologist is most likely to calculate Z scores in order to a. compare the skew of two different distribution b. determine the shape of a distribution score c. compare scores obtained from two different distributions d. determine how far above or below the median a score is
c. compare scores obtained from two different distributions *The Central Limit Theorem explains that regardless of the distribution scores in a population, the sampling distribution of sample means selected at random from that population will approach the shape of a normal distribution. Therefore we can convert any sampling distribution with any man and standard error to a standard normal distribution by applying the z transformation.
Which of the following will decrease the value of the test statistic in a one-sample t-test? a. increase the sample size b. reduce the level of significance from 0.05 to 0.01 c. sample variance is doubled d. the difference between the sample and population mean is increased
c. sample variance is doubled
Types of research
correlational experiemental
A researcher report concludes that there are significant differences among treatments, with "F(2, 27) = 8.62, p < 0.05." If the same number of participants was used in all of the treatment conditions, then how many individuals were in each treatment? a) 30 b) 27 c) 9 d) 10
d) 10
If the results of a two-factor experiment are presented in a line graph, then what pattern appears in the graph if there is an interaction? a) The line graph is not straight. b) The lines are parallel. c) The lines are separated by a space. d) The lines move toward each other or cross.
d) The lines move toward each other or cross
Scores on SAT form a normal distribution with a mean of μ = 500 with σ = 100. If the state college only accepts students who score in the top 60% on the SAT, what is the minimum score needed to be accepted? a) X = 560 b) X = 440 c) X = 525 d) X = 475
d) X = 475
A random sample of n = 4 scores is obtained from a population with a mean of µ = 90 and a standard deviation of σ 10. If the sample mean M 110, what is the z-score for the sample mean? a. 20 b. 5 c. 2 d. 4
d. 4
In the two-way ANOVA table below looking at the effects of birth order and attachment style of self-esteem, there is evidence of a: a. An interaction effect b. A main effect of birth order c. A main effect of attachment style d. Both b and c
d. Both b and c
. If the null hypothesis is rejected, variability between groups reflects a. only the treatment effect b. only random error c. random error and possibly treatment effect d. both random error and the treatment effect
d. both random error and the treatment effect
When researchers make various assumptions about the nature of the population from which the sample involved is drawn, this is a requirement of a. nonparametric techniques. b. the null hypothesis. c. sampling error. d. parametric techniques
d. parametric techniques *Parametric tests hypothesis tests that are used to test hypotheses about parameters in a population in which the data are normally distributed and measured on an interval or ratio scale of measurement
t test statistic
difference due to treaatment / difference due to error t= µ -M / Sm
9. Which of the following is true about a normal distribution. a. Has the same general bell shaped curve that is symmetric with respect to the vertical line that passes through the peak of the curve. b. The area under the curve is always 1 c. The curve is completely determined by the mean, and standard deviation. d. The empirical rule (68%, 95%, 99.7%) for mound shaped data applies to variables with normal distributions e. All of the above
e. All of the above
Experimental Research
manipulate a variable and then make observations
Degrees of Freedom
n-1
degrees of freedom for t test statistic
n-1. (number of sample size subtracted by 1. ex: n=9 df=n- 1 = 9-1
Correlational research
observations of what we see
Standard deviations of a sample
or S= square root of SS/ n-1
standard error of mean
r(m) r/ sq.r. n
t tests: Varince
s^2= sSS/df
Type 1 error
sample data shows a treatment effect when there isnt one. Is equal to the alpha level
Type 2 errors
sample does not appear to show that it was affected by the treatment when it actually has been. usually has a small treatment effect
Standard deviation of a population-parameter σ Unbiased
square root of ss/N
Standard Distribution population
z= X- μ/ σ σ = square root of ss/n set score of 6 (Set of 6 scores (0, 6, 5, 2, 3, 2) µ=18/6=3 and a SD=2
Z score
z= x- μ/ σ - The numerator is the deviation score (distance in points between X and μ and if above or below mean) - Deviation score is then divided by σ because we want the z-score to measure distance in terms of standard deviation unit.
Standard dist. Sample
z= x-M/ s s= square root of SS/n-1 The samples of z-scores will have the same shape as the original scores The sample of z-scores will have a M=0 The sample of z-scores will have a standard deviation of s=1
What is the minimum score to get in the top %15 of the SAT distribution?
µ = 500 σ=100 Top 15%=.15 1. Find the z-score that corresponds with proportion (column C)From table value =0.1492 2.Corresponding z-score z=1.04 3. Change z-score into X value 1.04σ=1.04(100)=104 points µ=500+104=604 points 4.Your score must be 604 points to be in the top 15% of the SAT
Standard deviation µ or M
µ:Standard deviation for the population or sample... -Parameters for populations -Statistics for samples M: Samples are a good estimate, but... -Produces a biased result -Tends to ignore extreme scores Biased statistic Any average value that either under or over estimates the population parameter
alpha values
1.z= 1.96 & -1.96 2. z= 2.56 &-2.56 3. z=3.30. &-3.30
. In general, the mean height of women is 65″ with a standard deviation of 3.5″. What is the probability of finding a random sample of 50 women with a mean height of 70″, assuming the heights are normally distributed?
Answer z = (x - μ) / (σ / √n) = (70 - 65) / (3.5/√50) 5 / 0.4 5 10.1
How does calculating the sample variance differ from calculating the population variance?
Answer: Population Sample
A distribution is normal if it has the correct proportions
Both sides are the same +/-1sd = 68.26% +/-2sd = 95.5% +/-3sd = 99.7%
Standard Deviation: Sum of Squares Definitional and Computional
Definitional SS=∑(X-µ)2 SS=40 Computational SS+∑X^2 - (∑X)^2/N
reate a study or give an example of one that would require you to use a a. One - Way ANOVA ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ b. Chi- Square ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ c. One- Sample t-test ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________
Chi - Square Example: Suppose you conducted a drug trial on a group of animals and you hypothesized that the animals receiving the drug would show increased heart rates compared to those that did not receive the drug. You conduct the study and collect the following data: Ho: The proportion of animals whose heart rate increased is independent of drug treatment. Ha: The proportion of animals whose heart rate increased is associated with drug treatment. Applying the formula above we get: Chi square = 105[(36)(25) - (14)(30)]2 / (50)(55)(39)(66) = 3.418 Before we can proceed we eed to know how many degrees of freedom we have. When a comparison is made between one sample and another, a simple rule is that the degrees of freedom equal (number of columns minus one) x (number of rows minus one) not counting the totals for rows or columns. For our data this gives (2-1) x (2-1) = 1. We now have our chi square statistic (x2 = 3.418), our predetermined alpha level of significance (0.05), and our degrees of freedom (df = 1). Entering the Chi square distribution table with 1 degree of freedom and reading along the row we find our value of x2 (3.418) lies between 2.706 and 3.841. The corresponding probability is between the 0.10 and 0.05 probability levels. That means that the p-value is above 0.05 (it is actually 0.065). Since a p-value of 0.65 is greater than the conventionally accepted significance level of 0.05 (i.e. p > 0.05) we fail to reject the null hypothesis. In other words, there is no statistically significant difference in the proportion of animals whose heart rate increased. 30. CORRELATION DOES NOT EQUAL CAUSATION! For example it might be significant that there is a correlation between ice cream sales and drowning, but that does not mean that eating ice cream causes people to drown. It just so
Dependent variable
The observed variable/behavior The dependent DEPENDS on the independent!
Quantitative variable
Varies by amount Usually measured in numbers
Qualitative variable
Varies by class Physical attributes
9. A distribution of examination scores yields the following information: Mean = 78 Mode = 79 Median =83 Based on these measures of central tendency, which of the following best describes the likely shape of the distribution? a. Bimodal b. normal c. positively skewed d. negatively skewed
d. negatively skewed
If our first exam has m = 68 points and s = 6 points. What is p(X>80)?
z = 80-68/8 = 2.00 sd The score of X=80 is exactly 2 standard deviations above the mean (z-score = +2.00)
z test statistic
z= M-mu / r (m)
If study results were: n=25. M=92 r= 20. SEM=4
z= M-mu/r (m) ; 92-80/4 = 3.00