STAT FINAL
The following ranges are possible 95% confidence intervals for the percentage of Americans who think that recent changes in the stock market have made them feel less confident about the nation's economy. For which of the confidence intervals could you conclude that a majority of the population feels that recent changes in the stock market have made them feel less confident about the nation's economy?
(.52, .55) Majority means more than 50%. For a confidence interval to support the statement of 'majority', then entire CI would need to be greater than .50.
A firm wishes to compare four programs for training workers to perform a certain manual task. Twenty new employees are randomly assigned to the training programs, with 5 in each program. At the end of the training period, a test is conducted to see how quickly trainees can perform the task. The number of times the task is performed per minute is recorded for each trainee.
1-way ANOVA
A test preparation company claims that more than 60% of the people who take their GRE prep course improve their score by at least 10 points. They conduct a 1 proportion hypothesis test using Ho: p=0.60 and Ha: p>0.60 and obtain a P-value of 0.39. Which of the following is a correct interpretation of this P-value? (Assume that the process was conducted appropriately.)
1. Assuming the true proportion equals 0.60, there is a 0.39 probability of getting a sample proportion at least as extreme as the one they got from sampling. 2.About 39 in 100 samples will give a sample proportion as high or higher than the one obtained if the population proportion really is 0.6. 3. If 60% of the people who take the prep course improve their score by at least 10 points, about 390 out of every 1000 random samples of the same size would produce the same result observed in this study or a result more unusual.
Assumptions of ANOVA
1. each of the k pop. or treatment response distributions is normal. 2. the observations in the sample from any particular one of the k populations are independent of each other. 3. the standard deviations of the k distributions are not significantly different.
Assumption for the z confidence interval for a pop. proportion
1. the sample is a random sample from the pop. of interest. 2. The sample size is large enough for the sampling dist. of p to be approx. normal. 3. sampling is without replacement
How many types of degrees of freedom characterize an F distribution?
2 In an F-test, there are two degrees of freedom--one from the regression row in ANOVA (k) and one from the error row (n-1-k).
A random sample of 8 new SUVs and another sample of 8 mid-size cars are tested for front impact resistance. The amounts of damage (in hundreds of dollars) to the vehicles when crashed at 20 mph head on into a stationary barrier are recorded. Investigators want to know if the SUV's are sturdier and thus suffer less damage.
2 sample t-test
According to a publication, college-bound males have out-performed college-bound females on the mathematics portion of tests given by the ACT Program. Samples of this year's scores have been collected. Does it appear that college-bound males are, on average, still outperforming college bound females on the mathematics portion of ACT tests?
2 sample t-test
Independent random samples of 17 sophomores and 13 juniors attending a large university were taken and grade point averages (GPAs) collected. At the 5% significance level, do the data provide sufficient evidence to conclude that the mean GPAs of sophomores and juniors at the university differ?
2 sample t-test
A recent survey asked San Diego residents the following question: "If you had been aware that before Schwarzenegger ran for Governor, he had fathered a child outside of his marriage, would you still have voted for him?" Out of the 136 males surveyed who had voted for Schwarzenegger at least once 78 said they would still vote for him, and out of the 128 females 48 said they would. Do these data provide convincing evidence to suggest that there is a difference between the proportion of male and female San Diego residents who would have voted for Schwarzenegger had they known he had fathered a child outside of his marriage?
2- proportion
The effective life (in hours) of batteries is compared by material type (1, 2 or 3) and operating temperature: Low (-10˚C), Medium (20˚C) or High (45˚C). Twelve batteries are randomly selected from each material type and are then randomly allocated to each temperature level. The resulting life of all 36 batteries is shown below. Is there difference in the mean life of the batteries for the different material type and different operating temperature levels?
2-way ANOVA
In a Test of Independence how many degrees of freedom does the chi-square distribution have for a table with 4 columns and 3 rows?
6 degrees of freedom are found using this formula df = (r - 1)(c - 1) where r stands for the number of rows and c stands for the number of columns in a two-way table. df = (4 - 1)(3 - 1) = (3)(2) = 6 Note: One variable has 4 levels (therefore 4 columns in the table) and one variable has 3 levels (therefore 3 rows in the the table)
Which of the following statements is correct about a Paired T-test?
A paired t-test is a one-sample t-test on the differences of the observations in the two groups
An experiment was designed to study the performance of four different detergents in cleaning clothes. The following "cleanliness" readings [higher = cleaner] were obtained with specially designed equipment for three different types of common stains. Use "stains" as the block variable. Is there a difference between the detergents?
ANOVA with block variable
What is another name for the 0-1 variable that is often used to represent a binary categorical variable in a multiple regression model?
Both a dummy variable and an indicator variable are other names for the 0-1 variable.
Assuming all assumptions have been met, which test would be used to study these data? Is the distribution of living conditions the same for all year levels of college students? Suppose that multiple samples of data were collected from second, third and fourth year college students. Each sample was asked to indicate their living conditions: Dorm, Apartment, With Parents, Other.
Chi-Square Test for Homogeneity When multiple samples are collected and the same categorical variable is compared across the samples, this is a test for homogeneity. The multiple samples were taken from the different college year levels and the categorical variable is the living condition.
Recently 195 North Georgia students were given a perfectionism survey. The chart below compares whether or not they are a perfectionist with their gender. Test at the .05 level of significance whether there is a relationship between gender and perfectionism.
Chi-square test of independence
Cuckoos lay their eggs in the nests of other (host) birds. The eggs are then adopted and hatched by the host birds, but the potential host birds lay eggs of different sizes. A random sample of sparrow host eggs and wagtail host eggs was taken and the length of the cuckoo eggs for each host was recorded. Based on the sample data, suppose a 95% confidence interval for the difference in mean lengths of cuckoo eggs (sparrow hosts minus wagtail hosts) is (- 0.6, - 0.1) mm. At the 5% significance level can you conclude that the size of the cuckoo eggs changes between sparrow and wagtail hosts, on average?
Cuckoos do change the size of their eggs between sparrow and wagtail hosts, on average, since 0 is not between the lower and upper bounds of the confidence interval.
What is the guideline for the assumption of a large sample when using a test statistic with a Chi-square distribution?
Every cell should have an expected count greater than or equal to 1 and no more than 20% of the expected cell counts are less than 5.
A(n) ______ test is an inferential procedure used to determine whether a frequency distribution follows a defined distribution.
Goodness of Fit The Chi-Square Goodness of Fit test compares observed counts to a proportional breakdown (either assumed equal or specific proportions given in the problem).
The director of a state agency believes that the average starting salary for clerical employees in the state is less than $30,000 per year. To test her hypothesis, she has collected a simple random sample of 100 starting clerical salaries from across the state and found that the sample mean is $29,750. State the appropriate null and alternative hypotheses.
H0 : μ = 30,000 Ha : μ < 30,000
A mail-order business prides itself in its ability to fill customers' orders in six calendar days or less, on average. Periodically, the operations manager selects a random sample of customer orders and determines the number of days required to fill the orders. Based on this sample information, he decides if the desired standard is not being met. He will assume that the average number of days to fill customers' orders is six days or less unless the data suggest strongly otherwise. Establish the appropriate null and alternative hypotheses.
H0 : μ = 6 days Ha : μ > 6 days
What is the FIRST null hypothesis (Ho) to study for a two-factor analysis of variance?
Ho: There is no significant interaction between factors
Suppose a study is conducted to determine if customers at grocery stores purchased the four major brands of cereal in the same proportion. What would be the null hypothesis for analyzing this categorical variable? Assume the grocery store only sells these four brands of cereal.
Ho: p1 = 0.25, p2 = 0.25, p3 = 0.25, p4 = 0.25 With four brands of cereals, you are studying a categorical variable with four levels. We will assume equal proportions, therefore p1=p2=p3=p4 = ¼.
Which of the following represents the null and alternative hypotheses for the model utility (significance) test for simple linear regression?
Ho: β = 0 versus Ha: β ≠ 0 In simple linear regression, we will assume the coefficient is zero in the null and look for evidence that it is not zero in the alternative.
How do you determine that a Chi-Square problem is a test of Independence (Association) and not a test of Homogeneity of proportions or a Goodness of Fit Test?
If you have one sample that is later categorized by the levels of two variables, then it is a test of independence. A test of independence (test of association) is used when one sample of data is gathered and two categorical variables are studied.
How do you determine that a Chi-Square problem is a test of Homogeneity and not a test of Independence (Association) or a Goodness of Fit Test?
If you multiple independent samples have been taken, and then the proportions displaying some quality are compared, then it is a test of homogeneity. Yes, this defines homogeneity. Multiple samples taken and one categorical variable being compared across the samples.
Suppose a 95% confidence interval for the true proportion of Americans who exercise regularly is (0.29, 0.37) Which one of the following statements is FALSE?
It is reasonable to say that a majority of Americans exercise regularly.
Suppose we are interested in how the exercise and body mass index affect the blood pressure. A random sample of 10 males is selected and their body mass index (BMI), number of hours of exercise per week, and the blood pressure are measured.
Multiple linear regression
A mining company is interested in the pH levels of soil before and after mining is completed. The pH levels from 15 areas of the mining land are taken before mining begins. At the conclusion of mining, the land is restored and another set of pH-readings were taken on the same 15 grids. Are the pH levels statistically different, on average?
Paired
Eleven tires were measured for tread wear using two different methods. In other words, each tire was reviewed twice - once using a weight-based method and second, for the same tire, using the typical method that is based on groove wear. Does it appear the two methods, on average, give different results?
Paired t-test
Trace metals in drinking water affect the flavor and an unusually high concentration can pose a health hazard. Ten pairs of data were taken measuring zinc concentration in the bottom of a glass of water and on the surface of the same glass of water. Do the data suggest that the true average concentration in the bottom water exceeds that of surface water?
Paired t-test
A __________ is an experimental design that controls for extraneous variation when comparing treatments.
Randomized block design
A 95% confidence interval and a 99% confidence interval are computed from the same set of data. Which of the following statements is correct?
The 99% confidence interval is wider
In a study, subjects are randomly assigned to one of three groups: control, experimental A, or experimental B. After treatment, the mean scores for the three groups are compared. The appropriate statistical test for comparing these means is...
The Analysis of Variance An ANOVA test is used to compare more than two means.
Which of the following is a property of the Chi-square distribution?
The Chi-square curve is skewed to the right. Yes, the chi-square distribution is skewed right (long tail on the right).
In regression, the residuals are which of the following?
The difference between the observed responses and the values predicted by the regression line.
which of the following statements is true with respect to the t-distribution?
The exact shape of the t-distribution depends on the number of degrees of freedom.
In least squares regression, which of the following is not a required assumption about the error term, ε?
The expected value of the error term is one.
Which of the following is not true with regards to a p-value?
The larger the p-value, the more conclusive the evidence is against the null hypothesis.
What is the interpretation of the null hypothesis Ho: β2 = 0 for the model y = α + β1x1 + β2x2 +β3x3 + ε ?
The null hypothesis states that as long as the other predictors x1 and x3 remain in the model, the predictor x2 provides no useful information about y. In a multiple linear regression model, when you are testing the coefficient of one of your predictors, it is under the assumption that the other predictors are still in the model. If the coefficient of predictor 2, β, is equal to zero, then this would mean that predictor 2 is not useful.
Which of the following is a basic assumption of the simple linear regression model?
The standard deviation (spread) of ε is the same for any particular value of x. ε, epsilon, represents the random error term or residual. One assumption of a linear model is that the residuals have constant spread or constant error variance. This is another way of saying that the standard deviation (spread) is the same/constant.
when n>30 in a one-sample means test...
This means that the sampling distribution of the sample means is approx. normally distributed therefore the normality assumption has been satisfied.
What is the function of a post-test (Tukey's Test) in ANOVA?
To determine which pairwise means are statistically different. Once we have determined that we have a significant ANOVA, we will continue on to use Tukey's pair-wise comparisons to determine which means are statistically different and which are not.
What does it mean to say that two factors interact?
Two factors interact if the average change in response associated with changing the level of one factor depends on the level of the other factor.
A 1975 article reported on the results of trials in which clouds were seeded and the amount of rainfall was recorded. The authors reported on the amount of rainfall from 26 seeded and 26 unseeded clouds. Which test should be used to determine if a difference exists in average rainfall between seeded and unseeded clouds?
Two-sample t test
The real estate industry claims that it is the best and most effective system to market residential real estate. A survey of randomly selected home sellers in Illinois found that a 95% confidence interval for the proportion of homes that are sold by a real estate agent is 69% to 81%. Interpret the interval in this context.
We are 95% confident that the true proportion of homes in Illinois sold by a real estate agent is between 69% and 81%.
When should a paired t-test be performed?
When the variable of interest is numerical, there are two groups being compared, and the samples taken are dependent.
Which of the following is the correct test statistic for analyzing multiple samples of categorical variables with many levels?
X2 When studying multiple samples of categorical data, you will use a Chi-Square test!
A confidence interval estimate is a range of values used to estimate
a population parameter
A regression analysis between sales (in $1000) and price (in dollars) resulted in the following equation: salesˆ=50,000−8(price)
an increase of $1 in price is associated with a decrease of $8000 in sales.
The American Pet Products Association conducted a survey in 2011 and determined that 60% of dog owners have only one dog, 28% have two dogs, and 12% have three or more. Supposing that you have decided to conduct your own survey and have collected the data below, determine whether your data supports the results of the APPA study. Perform the appropriate test using these data.
chi-square goodness of fit
Time magazine conducted a sample of American smokers and a sample of American non-smokers. The question posed of each group was: "Should the federal tax on cigarettes be raised to pay for health care reform?" Respondents chose between the following responses: Yes, No, Undecided. Is there sufficient evidence to conclude that the two populations — smokers and non-smokers — differ significantly with respect to their opinions?
chi-square test for homogeneity
The __________ measures the extent to which the x and y values are linearly related.
correlation coefficient (r) A measure of the strength and direction of the linear relationship between two numerical variables is Pearson's Correlation value.
If the coefficient of determination is a positive value, then the regression equation...
could have either a positive or negative slope.
How many degrees of freedom is the t critical value based on when computing a confidence interval for beta in a multiple regression model with four independent predictor variables if the sample size is 15?
df = 10 Hide Feedback When constructing a CI about beta (or conducting the test of significant regression) the degrees of freedom come from the error row of ANOVA. To calculate the degrees of freedom, use n - 1- k (where k represents the number of predictors in the model. 15 -1 - 4 = 10
As the test statistic becomes larger, the p-value
gets smaller
A P-value that is much smaller than the level of significance...
indicates evidence in support of the alternative hypothesis.
The Tukey-Kramer procedure...
is used to test for difference in pairwise means if, in the original hypothesis test for main effect, the p-value is less than the level of significance.
In simple linear regression, if the Pearson correlation value is a positive value, then the slope of the regression line...
must also be positive. If the correlation value is positive, then the slope of the line is positive. if the correlation value is negative, then the slope of the line is negative.
the p-value...
must be a number between 0 and 1
which of the following is not true
paired samples are also independent samples
A __________ is a number, based on sample data that represents a plausible value of a population characteristic.
point estimate
The difference between the observed and predicted value of the response variable is a...
residual A residual is calculated by taking the observed y - predicted y.
In regression analysis, the variable that is being predicted is the...
response, or dependent, variable
For a sample of n=17 pupils in a particular school, we might be interested in the relationship of two variables as follows: Exam performance at age 16, Exam performance at age 11. Can the exam performance at age 16 be predicted given the exam performance at age 11?
simple linear model
To conduct an analysis of variance (ANOVA) on a randomized complete block design a researcher would use...
the ANOVA General Linear Model and use two categorical variables as the factors (one factor and one block variable). Both the factor you are interested in and the block variable will be used as factors in ANOVA.
What happens to the value of the F-statistic in an analysis of variance when the treatment mean square (MSTr) gets larger but the error mean square (MSE) remains constant? F= MSTR/MSE
the F test statistic will get larger
When performing a hypothesis test upon two dependent samples of numerical data, the variable of interest is...
the differences that exist between the matched-pair data.
Levene's test is an inferential method that is used to test...
the equality of three or more population variances
The ANOVA procedure is a statistical approach for determining whether or not...
the means of two or more populations are equal
The Pearson correlation value is used to determine...
the strength and direction of the linear relationship between the predictor variable and the response variable.
In the analysis of variance procedure (ANOVA), "factor" refers to...
the treatment (that is categorical and can have many levels)
The required condition for using an ANOVA procedure on data from several populations is that the...
the variance must be constant for each population
If we change a 95% confidence interval estimate to a 99% confidence interval estimate, we can expect...
the width of the confidence interval to increase
In a χ2 test of independence, the null hypothesis is that...
there is not an association between two variables.
A Goodness of Fit Test is always conducted as a...
upper-tail test The Chi-square distribution is a skewed right distribution and therefore a right tailed test
Data collected by child development scientists produced the following 90% confidence interval for the average age (in months) at which children say their first word: (10.4 months, 13.8 months). Which of the following statements is a correct interpretation of the confidence interval.
we can say, with 90% confidence that the true mean age at which children say their first word is between 10.4 and 13.8 months
To determine whether the NPP indicates that the sample data could have come from a population that is normally distributed
we look for a p-value which is greater than 0.05
When should a paired t-test be performed instead of a two-sample t-test?
when each observation in one group has a dependence on a particular observation in the other group
Which of the following is the correct test statistic to use for analyzing the difference between 2 proportions from 2 independent samples?
z When you compare proportions from two independent samples you will conduct a 2-proportions z test.
Which of the following is the correct alternative hypothesis (Ha) for testing whether two population means are the same?
μ1 - μ2 ≠ 0
