Deck 18: Review of Statistical Inference

Question 1

The test statistic used to test hypotheses about the population variance is given by $\chi$ ² = (n -1)s² / $\sigma$ ², which is chi-squared distributed with n - 1 degrees of freedom when the population is normally distributed with variance equal to $\sigma$ ².

Answer 1

True

Question 2

The expected value of is .

Answer 2

True

Question 3

Two uses for the population variance are to measure risk and consistency.

Answer 3

True

Question 4

We use a t-test to determine whether two population variances are equal.

Question 5

The sampling distribution of is normal if the sampled populations are normal, and approximately normal if the populations are nonnormal and the sample sizes n₁ and n₂ are large.

Question 6

The statistic when the sampled population is normal is Student t-distributed with n degrees of freedom.

Question 7

The pooled-variances t-test requires that the two population variances need not be the same.

Question 8

The sum of squares for treatments, SST, achieves its smallest value (zero) when all the sample means are equal.

Question 9

The F-distribution is symmetric.

Question 10

In estimating the population mean with the population standard deviation unknown, if the sample size is 16, there are 8 degrees of freedom.

Question 11

We use the analysis of variance (ANOVA) technique to compare two or more population means.

Question 12

When comparing two population variances, we use the ratio rather than the difference .

Question 13

The test statistic employed to test is is F-distributed with v₁ = n₁ - 1 and v₂ = n₂ - 1 degrees of freedom if the two populations are F-distributed.

Question 14

The t-distribution is used in a confidence interval for a mean when the actual standard error is not known.

Question 15

Independent samples are those for which the selection process for one is not related to the selection process for the other.

Question 16

In order to determine the p-value associated with hypothesis testing about the population mean $\mu$ , it is necessary to know the value of the test statistic.

Question 17

The F-test used for testing the difference in 2 population variances is always a one-tailed test.

Question 18

If the sampled population is nonnormal, the t-test of the population mean $\mu$ is still valid, provided that the condition is not extreme.

Question 19

A condition of using the chi-squared test statistic for testing the variance is that the population random variable have a normal distribution.

Question 20

The F-test used in one-way ANOVA is an extension of the t-test of $\mu$ ₁ - $\mu$ ₂.

Question 21

In a test of a contingency table, rejecting the null hypothesis concludes the variables are not independent.

Question 22

The alternative hypothesis in a goodness-of-fit test is that none of the p_i values are equal to their values specified in H₀.

Question 23

The degrees of freedom for the test statistic in a test of a contingency table is (r - 1)(c - 1) where r is the number of rows in the table, and c is the number of columns.

Question 24

A chi-squared goodness-of-fit test is always a two-tailed test.

Question 25

If two events A and B are independent, the P(A and B) = P(A) + P(B).

Question 26

The degrees of freedom for the test statistic in a test of a contingency table is (r - 1)(c -1) where r is the number of rows in the table, and c is the number of columns.

Question 27

The test statistic for the chi-squared test of a contingency table is the same as the test statistic for the goodness-of-fit test.

Question 28

Conducting t-tests for each pair or population means is statistically equivalent to conducting one F-test comparing all the population means.

Question 29

The Bonferroni adjustment to Fisher's Least Significant Difference (LSD) multiple comparison method is made by dividing the specified experimentwise Type I error rate by the number of pairs of population means.

Question 30

Multiple comparison methods are used to determine whether or not any differences occur amongst a group of population means.

Question 31

If the expected frequency of a cell is less than 5, you should combine cells of the table.

Question 32

One example of a blocking variable is the dosage level that each subject is assigned to in a randomized experiment.

Question 33

In conducting a chi-squared goodness-of-fit test, an essential condition is that all expected frequencies are at least five.

Question 34

In one-way ANOVA, the test statistic is defined as the ratio of the mean square for error (MSE) and the mean square for treatments (MST), namely, F = MSE / MST.

Question 35

In employing the randomized block design, the primary interest lies in reducing sum of squares for blocks (SSB).

Question 36

A chi-squared test is used to describe a population of nominal data.

Question 37

If the numerator (MST) degrees of freedom is 3 and the denominator (MSE) degrees of freedom is 18, the total number of observations must equal 21.

Question 38

A multinomial experiment with two categories is identical to a binomial experiment.

Question 39

In a chi-squared test of a contingency table, the value of the test statistic was $\chi$ ² = 15.652, and the critical value at $\alpha$ = 0.025 was 11.1433. Thus, we must reject the null hypothesis at $\alpha$ = 0.025.

Question 40

Tukey's multiple comparison method is more powerful than Fisher's LSD Method at finding differences in pairwise population means.

Question 41

A regression analysis between sales (in $) and advertising (in $) resulted in the following least squares line: . This implies that an increase of $1 in advertising is associated with an increase of $60 in sales.

Question 42

A confidence interval (as opposed to a prediction interval) is used to estimate the long-run average value of y.

Question 43

In multiple regression analysis, the adjusted coefficient of determination is adjusted for the number of independent variables and the sample size.

Question 44

If the coefficient of correlation is 1.0, then the coefficient of determination must be 1.0.

Question 45

The coefficient of determination is equal to the coefficient of correlation squared.

Question 46

A zero population correlation coefficient for x and y means that there is no type of relationship whatsoever between x and y.

Question 47

The residual r_i is defined as the difference between the actual value y_i and the estimated value .

Question 48

In simple linear regression, the denominator of the standard error of estimate s _{$\varepsilon$} is .

Question 49

In reference to the equation , the value 0.12 is the average change in y per unit change in x₁, when x₂ is held constant.

Question 50

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be 1.0.

Question 51

In testing the significance of a multiple regression model with three independent variables, the null hypothesis is .

Question 52

The value of the sum of squares for regression SSR can never be smaller than 0.0.

Question 53

One method of diagnosing heteroscedasticity is to plot the residuals against the predicted values of y, then look for a change in the spread of the plotted values.

Question 54

Statisticians have shown that sample y-intercept b₀ and sample slope coefficient b₁ are unbiased estimators of the population regression parameters $\beta$ ₀ and $\beta$ ₁, respectively.

Question 55

Correlation analysis is used to determine whether there is a linear relationship between an independent variable x and a dependent variable y.

Question 56

Another name for the residual term in a regression equation is random error.

Question 57

The coefficient of determination R² measures the proportion of variation in y that is explained by the explanatory variables included in the model.

Question 58

An inverse relationship between an independent variable x and a dependent variably y means that as x increases, y decreases, and vice versa.

Question 59

Most statistical software print a second R² statistic, called the coefficient of determination adjusted for degrees of freedom, which has been adjusted to take into account the sample size and the number of independent variables.

Question 60

A prediction interval is used when we want to predict a one-time occurrence for a particular value of y when the independent variable is a given x value.

Question 61

A random sample of 25 observations is selected from a normally distributed population. The sample variance is 10. In the 95% confidence interval for the population variance, the upper limit is:

A) 19.353
B) 17.331
C) 17.110
D) 6.097

Question 62

In testing for the differences between the means of two independent populations where the variances in each population are unknown but assumed equal, the degrees of freedom is:

A) n₁ + n₂
B) n₁ + n₂ - 2
C) n₁ + n₂ - 1
D) None of these choices

Question 63

Two independent samples of sizes 20 and 30 are randomly selected from two normally distributed populations. Assume that the population variances are unknown but equal. In order to test the difference between the population means, , the sampling distribution of the sample mean difference, , is:

A) normal.
B) Student-t with 50 degrees of freedom.
C) Student-t with 48 degrees of freedom.
D) None of these choices.

Question 64

When testing vs. , the observed value of the z-score was found to be -2.15. Then, the p-value for this test would be

A) .0158
B) .0316
C) .9842
D) .9684

Question 65

From the coefficient of determination, we cannot detect the strength of the relationship between the dependent variable y and any individual independent variable.

Question 66

Based on sample data, the 90% confidence interval limits for the population mean are LCL = 170.86 and UCL = 195.42. If the 10% level of significance were used in testing the hypotheses H₀: $\mu$ = 201 vs. H₁: $\mu$ $\neq$ 201, the null hypothesis:

A) would be rejected.
B) would be accepted.
C) would fail to be rejected.
D) would become H₀: $\mu$$\neq$ 201

Question 67

Which of the following conditions is needed regarding the chi-squared test statistic for the test of variance?

A) The population random variable must be normal.
B) The test statistic must be a non-negative number.
C) The test statistic must have a chi-squared distribution with n - 1 degrees of freedom.
D) All of these choices are true.

Question 68

Multicollinearity is present if the dependent variable is linearly related to one of the explanatory variables.

Question 69

In calculating the standard error of the estimate, , there are (n - k - 1) degrees of freedom, where n is the sample size and k is the number of independent variables in the model.

Question 70

The t-test for the difference between the means of two independent populations assumes that the respective:

A) sample sizes are equal.
B) populations are normal.
C) means are equal.
D) All of these choices are true.

Question 71

For statistical inference about the mean of a single population when the population standard deviation is unknown, the degrees for freedom for the t-distribution equal n - 1 because we lose one degree of freedom by using the:

A) sample mean as an estimate of the population mean.
B) sample standard deviation as an estimate of the population standard deviation.
C) sample proportion as an estimate of the population proportion.
D) sample size as an estimate of the population size.

Question 72

The statistic (n - 1)s² / $\sigma$ ² has a chi-squared distribution with n - 1 degrees of freedom if:

A) the sample has a Student t-distribution with degrees of freedom equal to n -1.
B) the sample is normally distributed with variance equal to s².
C) the population is normally distributed with variance equal to $\sigma$ ².
D) All of these choices are true.

Question 73

The F-distribution is the sampling distribution of the ratio of:

A) two sample variances.
B) two normal population means.
C) two normal population variances.
D) None of these choices.

Question 74

The expected value of the difference of two sample means equals the difference of the corresponding population means when:

A) the populations are normally distributed.
B) the samples are independent.
C) the populations are approximately normal and the sample sizes are large.
D) All of these choices are true.

Question 75

Which of the following is not an example illustrating the use of variance?

A) As a measure of risk.
B) As a judge of consistency.
C) To search for and reduce variability in a process.
D) All of these choices are true.

Question 76

Multicollinearity is a situation in which two or more of the independent variables are highly correlated with each other.

Question 77

A test is being conducted to test the difference between two population means using data that are gathered from a matched pairs experiment. If the paired differences are normal, then the distribution used for testing is the:

A) normal distribution.
B) binomial distribution.
C) Student t-distribution.
D) F-distribution.

Question 78

A random sample of size 15 taken from a normally distributed population revealed a sample mean of 75 and a sample variance of 25. The upper limit of a 95% confidence interval for the population mean would equal:

A) 77.77
B) 72.23
C) 88.85
D) 77.27

Question 79

For a 99% confidence interval of the population mean based on a sample of n = 25 with s = 0.05, the critical value of t is:

A) 2.7969
B) 2.7874
C) 2.4922
D) 2.4851

Question 80

Multicollinearity affects the t-tests of the individual coefficients as well as the F-test in the analysis of variance for regression because the F-test combines the t-tests into a single test.