Module 11

Chi square critical value calculation

(row - 1)(column-1)

Which of the following statements are true about the confidence intervals for sample correlation coefficient s? A). 95% of all sampling units will be found within the interval B) we are 95% confident that a randomly selected sampling unit be found within the interval C) 95% of the confidence intervals from repeated sampling will include the true population value D) 95% of all samples have mean values given by the interval

C

Shapiro-Wills test p > 0.05

Fail to reject and conclude that there is not enough evidence to state that the sample deviate from a normal distribution

Chi square tests are two tailed (TF)

False

Chi-squared distributions can be used to test whether the slope in a regression is different from zero.(TF)

False

F distributions can be used to test whether the counts in a contingency table are independent. (TF)

False

F distributions can be used to test whether the difference between two sample means is different from zero. (TF)

False

In any F distribution, roughly 95% of the probability distribution lies within ±2 standard deviations of the mean. (TF)

False

Normal distributions only have positive probability density when the X value (horizontal axis value) is positive. (TF)

False

The mean of any F distribution is equal to the number of degrees of freedom for the distribution. (TF)

False

The mean of any chi-squared distribution is 1. (TF)

False

t distributions can be used to test whether the counts in a contingency table are independent. (TF)

False

One way ANOVA assumptions

Response variables within the K population follow a normal distribution and have equal variance, and samples associated with each population are randomly selected and are dependent of all other samples, r

Total variation

is the variation from each data point to the grand mean

Suppose that a One-way ANOVA is being performed to compare the means of 4 populations and that the sample sizes are 15, 17, 20, and 14. Determine the degrees of freedom for the F-statistic. (a) the degree of freedom of the numerator (enter as an integer) (b) the degree of freedom of the denominator (enter as an integer)

3, 62

The null hypothesis for a ANOVA test is

All the means across all groups is equal

The alternative hypothesis for a ANOVA test is

At least one mean in the groups is not equal

ANOVA stands for: a)addition of values b)analysis of variance c)add now or value after d)average of values e)average number of votes

B

Which of the following sum of squares measures the variability of the observed values of the response variables around their respective means A) group B) residual C) regression D) total

B

What would be the advantage statistically of only looking for significant differences in a subset of all the possible pair-wise comparisons? A. By only looking at some pairs, we decrease the number of statistical tests, and that decreases the power of our tests. B. By only looking at some pairs, we decrease the number of statistical tests, and that increases the power of our tests. C. By only looking at some pairs, we increase the number of statistical tests, and that decreases the power of our tests. D. By only looking at some pairs, we increase the number of statistical tests, and that increases the power of our tests.

B For every increase in the number of tests on the same data, you lose statistical power. Limiting the number of specific pair-wise comparisons will mean fewer tests, and so increased statistical power (more likely to reject the null hypothesis if there is a real difference).

The P value for an ANOVA F statistic is large if the sample: a) sizes are small b) standard deviations are equal c) standard deviations are small d) sizes are equal e) means are about equal

E

We anticipate a small P value for an ANOVA F statistic if the box plots for the samples are: a) wide and have similar medians b) wide and similarly located c) symmetrical d) identical e) narrow and located differently

E

If your F distribution is skewed to the right, this means your df is small (TF)

False

The sample distribution for a F-test can have positive and negative sample variance (TF)

False, you can't have a sample variance less than 0

Chi-square hypothesis

Ho = The rows and columns are independent or The observed frequency is not different from the expected frequency Ha = The rows and columns are not independent or The observed frequency is different from the expected frequency

Bartlett's test null and alternative hypothesis

Ho = all groups have equal residual variance Ha = At least two groups do not have equal residual variance

Shapiro-wills null and alternative hypothesis are

Ho = residuals are Normally distributed Ha = Residuals are not Normally distributied

residual variation

The distance from our observed data points to the mean in each of the different groups

All F tests are one-tailed tests (TF)

True

As sample sizes get bigger, the F-distribution peak centres itself around the ratio of 1 (TF)

True

Chi-squared distributions can be used to test whether the counts in a contingency table are independent. (TF)

True

Chi-squared distributions only have positive probability density when the X value is positive. (TF)

True

F distributions can be used to test whether the slope in a regression is different from zero. (TF)

True

Sampling distribution of the sample correlation coefficient (r) follows a t-distribution (TF)

True

The mean of any F distribution if 1 (TF)

True

The mean of any chi-squared distribution is equal to the number of degrees of freedom for the distribution. (TF)

True

t distributions can be used to test whether the difference between two sample means is different from zero. (TF)

True

t distributions can be used to test whether the slope in a regression is different from zero. (TF)

True

To determine whether the test statistic of ANOVA is statically significant, it can be compared to a critical value. What two pieces of information are needed to determine the critical value? A) sample size, number of groups B) mean, sample standard deviation C) expected frequency, obtained frequency D) MSg, MSe

A

What are the three conditions required for one-way ANOVA? a) independent samples, normal populations and approximately equal population standard deviations b)large sample sizes, paired samples and normal differences c)normal populations, no outliers and equal sample standard deviations d)None of the above

A

Which of the following describes the sampling distribution of the sample correlation coefficient A) Histogram of sample correlation coefficients from a large number of repeated samples of a population B) Histogram of observed values from a large sample of the population C) Histogram of sample correlation coefficients from 100 repeated samples of a population D). Histogram of means from a large number of repeated samples of a population

A

Which of the following is true of the Tukey HSD test for pair-wise differences? A. The Tukey HSD test evaluates all possible pair-wise comparisons between levels, but may not report a large difference as significant if you have many levels in your data. B. The Tukey HSD test evaluates all possible pair-wise comparisons between levels, but may not report a large difference as significant if you have too few levels in your data. C. The Tukey HSD test evaluates the most important pair-wise comparisons between levels, but may not report a large difference as significant if you have many levels in your data. D. The Tukey HSD test evaluates the most important pair-wise comparisons between levels, but may not report a large difference as significant if you have too few levels in your data.

A The Tukey HSD test shows an evaluation of all possible pairs. E.g. for 4 groups/levels, we would have 3! = 6 pair-wise comparisons If we have more levels, then we would have more rows in that output, which means more tests against the null hypothesis and so higher odds of a "by chance" extreme test statistic. The Tukey HSD test tries to correct for this multiple testing by scaling up the p values more (making p less extreme) when there are more rows/pair-wise comparisons. This scaling leads to a lower power or lower chance of getting a significant p value on any individual pairwise comparison, which may then hide an interesting difference.

No free lunch: if we only look at some pair-wise comparisons (not all, like in the Tukey HSD), we will get higher power, and so lower p values every time. To be honest in our statistical reporting, which of the following must be true? A. We must have selected the subset of pair-wise comparisons before we collected our data, based on our hypothesis. B. We must have selected the subset of pair-wise comparisons after we collected our data, based on the between group differences. C. We must have selected the subset of pair-wise comparisons with the largest between-groups differences. D. We must have selected the subset of pair-wise comparisons with the smallest between-groups differences.

A Using fewer pair-wise tests means we get more extreme p values (so more likely significant p values)from the same data. However, we are trying not to fool ourselves into thinking an extreme p value is actually real just by chance, because our multiple tests make a "by chance" extreme value more likely. The only way to be sure we are not cherry-picking the most significant results out of all the pairwise comparisons is to have chosen our pairs of interest before the collection of any data. For the example in the video about animals recognizing shapes, to be "allowed" to only look at the "nutcracker vs. others" pairs and get the benefit of increased statistical power, we would have needed to make a prediction before the experiment like "nutcrackers will tend to have higher recognition abilities than all the other 3 species".

Which of the following studies would be most appropriately evaluated using an ANOVA F-test, as in the videos? A. Testing whether sex is independent of income category (high/middle/low). B. Testing whether sex is independent of adult height. C. Testing whether province of origin is independent of life expectancy. D. Testing whether province of origin is independent of job type (white collar, blue collar, or other).

C For an ANOVA F-test as in the video, the test is comparing a measured numerical value against a categorical value, usually with 3 or more levels. While an ANOVA can be used to compare the means of just two levels of a categorical variable (e.g. sex - male or female), usually a t-test with two independent samples is used in that case. (Note: R effectively does both an ANOVA F-test and a ttest when use the lm() command with only two levels.)

When conducting a pair-wise comparison between two groups/levels manually, we use the following formula for the standard error: SEp=s/s^2+1/n1+1/n2 What do the s2 and n values represent? A. s2 = the pooled variance within all groups; n1 = the total number of samples in the two specific groups; n2 = total samples in all the groups. B. s2 = the pooled variance within the two groups we are comparing; n1 = the total number of samples in the two specific groups; n2 = total samples in all the groups. C. s2 = the pooled variance within all groups; n1, n2 = the number of samples in the two specific groups we are comparing. D. s2 = the pooled variance within the two groups we are comparing; n1, n2 = the number of samples in the two specific groups we are comparing.

C When we estimate variance from a sample, we would like to use as much data as possible. Recall that our ANOVA assumptions (which you remembered to test with the Shapiro-Wills and Bartlett tests before getting this far!) include "variance within all the groups is the same". If that is true, to get the best variance estimate, we should use data from all the groups to estimate the sample variance. Once we have the sample variance though, and we need to scale the variance to get the standard deviation of the mean for null hypothesis, we are only talking about the means for those two groups, and that depends only on the number of samples in the two groups.

You are testing whether province of origin is independent of life expectancy using an ANOVA F-test, based on sample of 50 people each from each of the 10 provinces. How many degrees of freedom should used to construct the null hypothesis F distribution? A. df(10, 49) B. df(9, 49) C. df(10, 499) D. df(9, 490)

D The first degree of freedom is for the groups. If we are told the grand mean (overall mean), then we would get to choose the other 9 group means independently of that, so we have 9 degrees of freedom. The second degree of freedom is for the error or residuals. If we know all 10 of the group means, and have 500 people we measure (50 each from 10 provinces), there are 500-10 = 490 degrees of freedom for the raw data/residuals.

In a hospital with high cross-infection rates, the infectious disease group ran a trial with doctors and nurses assigned to one of four hand-washing protocols, and measured the number of bacteria on a swab taken at a random time for each person during the day. After recording the data and importing it into R, the team found the following results: the ANOVA F-test returned a p value of p = 1.72e-3; the Shapiro-Wills test for normality of the residuals returned a p value of p = 0.47; Barlett's test for equal variance returned a p value of p = 0.00242. What is the best interpretation of these results from the list below? A. We fail to reject the null hypothesis, and say there is no evidence from this study that different hand-washing techniques leave different numbers of bacteria on the hand. B. We reject the null hypothesis, and say there is evidence that different hand-washing techniques leave different numbers of bacteria on the hand. C. We have to discount the ANOVA F-test p value, because the data failed the Shapiro-Wills test for normality of the residuals. D. We have to discount the ANOVA F-test p value, because the data failed Bartlett's test for equal variance.

D The purpose of the Shapiro-Wills and Bartlett tests is to ensure that our data matches the assumptions that go into the ANOVA F-test null distribution. If you violate one of the assumptions, you really can't trust the ANOVA results. In the Shapiro-Wills and Bartlett tests, the null hypotheses are "my data is boring, and so meets the ANOVA assumptions". Getting a p = 0.00242 on the Bartlett test indicates "hey, my data aren't boring (specifically, the variances in all the groups are not equal), so I really can't trust the standard ANOVA results here."

ANOVA testing asks

Is the variation among group means greater than by chance along?

F distributions only have positive probability density when the F value (horizontal axis value) is positive. (TF)

True

Groups are not independent in an ANOVA test (TF)

True

In any normal distribution, roughly 95% of the probability distribution lies within ±2 standard deviations of the mean. (TF)

True

Group variation

The variation from the mean of each group to the grand mean (regression)

ANOVA has three sources of variation

Total, group and residual variation