One mean

Question: We want to investigate/estimate the mean of a population.

Example: What is the average length (in millimeters) of woolly bear caterpillars?

The majestic woolly bear caterpillar
The majestic woolly bear caterpillar

Parameter of Interest

  • \(\mu\) - the mean of all elements in the population (which is unknown)

Estimating \(\mu\) from a simple random sample

From a simple random sample drawn from the population we find three values:

  • \(n\) - the sample size
  • \(\overline{x}\) - the sample mean of the \(n\) values in the sample.
  • \(s\) - the standard deviation of the \(n\) values in the sample.

Confidence Interval for \(\mu\)

We use the formula \[\overline{x} \pm t^*\cdot\text{SE}\] where \[\text{SE}=\frac{s}{\sqrt{n}},\] and \(t^*\) is the value in the \(T\)-distribution with \(n-1\) degrees of freedom that corresponds to the desired confidence level.

Code to calculate the confidence interval in R

# the data in vector form
wb = c(52.0, 48.9, 47.3, 47.3, 48.6, 48.8, 42.8, 44.3, 46.7, 46.4, 
       47.5, 51.4, 49.0, 47.7, 48.5, 48.5, 45.8, 49.3, 48.1, 46.5) 
# summary statistics
n=length(wb) #sample size
xbar=mean(wb) #sample mean
s = sd(wb) #sample standard deviation
C = .95 # confidence level
df = n-1 #degrees of freedom
SE = s/sqrt(n)
tstar = qt(C + (1-C)/2,df)
MOE = tstar*SE #margin of error
xbar + c(-MOE,MOE) #returns confidence interval (low to high)
## [1] 46.78067 48.75933

Interpretation We are 95% confident that the population mean \(\mu\) length of woolly bear caterpillars is between 46.8 and 48.8 millimeters.

Remark 95% confidence does not refer to likelihood of individual caterpillar lengths landing in the interval - it is just making a claim about the average length of all woolly bear caterpillars. For instance, in our sample, only 8 of the 20 observations fall within the range of the interval.

Hypothesis Test

To test the null hypothesis \(H_o: \mu = \mu_0\) against one of three possible alternatives:

  • \(H_a: \mu > \mu_o\) (one-sided, greater than \(\mu_o\))
  • \(H_a: \mu < \mu_o\) (one-sided, less than \(\mu_o\))
  • \(H_a: \mu \neq \mu_0\) (two.sided)

Suppose we want to test the average length of all woolly bear caterpillars is 45 mm or not (perhaps a paper we read claimed the average length of wb caterpillars is 45 millimeters and we want to check whether that seems reasonable for wb caterpillars in our neck of the woods). We choose the hypotheses \[H_o: \mu = 45 ~~\text{against}~~H_a: \mu\neq 45.\]

Test statistic: This is a t-score computed under the assumption that \(H_o\) is true: \[t= \frac{\overline{x}-\mu_0}{\text{SE}}\] where \(\text{SE}=s/\sqrt{n}\).

Calculate the test statistic in R

# Value of mean under the null hypothesis
mu0 = 45 
## Data
wb = c(52.0, 48.9, 47.3, 47.3, 48.6, 48.8, 42.8, 44.3, 46.7, 46.4, 
       47.5, 51.4, 49.0, 47.7, 48.5, 48.5, 45.8, 49.3, 48.1, 46.5)
n=length(wb) #sample size
xbar=mean(wb) #sample mean
s = sd(wb) #sample standard deviation
# Standard error
SE = s/sqrt(n)
# Test statistic
t = (xbar-mu0)/SE
t
## [1] 5.860202

We can determine the p-value with the pt() function. How to use the pt() function depends on the alternative hypothesis.

In this case the p-value is

2*pt(-abs(t),n-1)
## [1] 1.209613e-05

Note: The p-value above is given in scientific notation: 1.21e-05 stands for \(1.21 \times 10^{-5} = 0.0000121\), which is, of course, a tiny p-value.

Conclusion This p-value is so small that we have convincing evidence that the null hypothesis is false. We have good reason to believe the average length of all woolly bear caterpillars is not 45 millimeters.

Using t.test()

The t.test() function is a black box r command that does confidence interval and test of significance calculations for us in one fell swoop for both a single population mean and for the difference of two means. (If you type t.test in the RStudio help search box you can get information about options.)

The following code reproduces the test of significance we just ran through, and it will also give the 95% confidence interval we calculated. Essentially we input the data vector, the value of the mean under the null hypothesis, and the nature of our alternative hypothesis (greater, less, or two.sided).

wb = c(52.0, 48.9, 47.3, 47.3, 48.6, 48.8, 42.8, 44.3, 46.7, 46.4, 
       47.5, 51.4, 49.0, 47.7, 48.5, 48.5, 45.8, 49.3, 48.1, 46.5)
t.test(wb, mu=45, alternative="two.sided")
## 
##  One Sample t-test
## 
## data:  wb
## t = 5.8602, df = 19, p-value = 1.21e-05
## alternative hypothesis: true mean is not equal to 45
## 95 percent confidence interval:
##  46.78067 48.75933
## sample estimates:
## mean of x 
##     47.77

Example: Does the sample recorded below under the name obs, drawn from a population with unknown mean \(\mu\), provide statistically significant evidence at the \(\alpha = .05\) level that \(\mu\) is less than 10?

obs <- c(10.1,9.4,9.8,8.6,10.2,9.1,10.0,8.9,9.4,9.7,10.3,9.6)
t.test(obs,mu=10,alternative="less")
## 
##  One Sample t-test
## 
## data:  obs
## t = -2.6521, df = 11, p-value = 0.01125
## alternative hypothesis: true mean is less than 10
## 95 percent confidence interval:
##      -Inf 9.868173
## sample estimates:
## mean of x 
##  9.591667

Conclusion The p-value of .01125 is less than 0.05. If the population mean were 10, the chances of gathering a random sample of size 12 having sample mean less than or equal to 9.59 is about 1.1% - this likelihood is small enough that we reject the null. We believe the population that this sample was drawn from has population mean less than 10.

One proportion

Question: We want to estimate/investigate the proportion of all elements in a population having a certain feature.

Example: Estimate the proportion of all Linfield Students who would pick 3 or 7 if asked to pick a random number between 1 and 10.

Parameter of Interest

  • \(p\) - proportion of all elements in the population having the feature. Generally, \(p\) is unknown, and our goal is to estimate it and/or make some claim about it.

Estimating \(p\)

From a simple random sample drawn from the population we find three values:

  • \(n\) - the sample size.
  • \(\hat{p}\) - the proportion of elements in the sample that have the feature. Note: \(\hat{p}=x/n\), where \(x\) is the number of successes, i.e., the number of observations having the feature of interest.

Confidence Interval for \(p\)

\[\hat{p} \pm z^* \cdot \text{SE}\] where \[SE \approx \sqrt{\frac{\hat{p}(1-\hat{p})}{n}},\] and \(z^*\) is the value in the standard normal distribution \(N(0,1)\) that corresponds to the desired confidence level.

Calculate the confidence interval in R

# Using class data from the 2021-22 academic year
n=86 #sample size
x=34 #successes 
C = .95 # confidence level
phat = x/n
SE = sqrt(phat*(1-phat)/n)
zstar = qnorm(C + (1-C)/2)
MOE = zstar*SE #margin of error
phat + c(-MOE,MOE) #Confidence Interval
## [1] 0.2920152 0.4986824

Interpretation: We are 95% confident that the proportion of all Linfield students who would pick 3 or 7 on the random number question is between .292 and .499.

Hypothesis test

To test the null hypothesis \[H_o: p = p_0\] against one of three possible alternatives:

  • \(H_a: p > p_o\) (one-sided, greater than \(p_o\))
  • \(H_a: p < p_o\) (one-sided, less than \(p_o\))
  • \(H_a: p \neq p_0\) (two.sided)

Test statistic. This is a z-score computed under the assumption that \(H_o\) is true: \[z = \frac{\hat{p}-p_o}{\text{SE}}\] where \[\text{SE}=\sqrt{\frac{p_o(1-p_o)}{n}}.\]

Suppose we want to test the claim that 20% of all Linfield students would pick 3 or 7 on the random number question (since 3 and 7 make up 2 of the possible 10 choices). We test the hypotheses \[H_o: p = 0.2 ~~\text{against}~~H_a: p \neq 0.2.\]

Calculate the test statistic in R

p0=.2
n=86 #sample size
x=34 #successes
phat = x/n
SE = sqrt(p0*(1-p0)/n) # If Ho is true, this is the standard error!!
z = (phat-p0)/SE
z
## [1] 4.528976

We can determine the p-value with the pnorm() function. How to use the pnorm() function depends on the alternative hypothesis.

In this case, since we have a 2-sided alternative, the p-value is

2*pnorm(-abs(z))
## [1] 5.92701e-06

Conclusion This p-value is so small that we reject the null and conclude that \(p \neq .2\). This conclusion agrees with the fact that our confidence interval above did not include .2 within it! We are confident that \(p \neq .2\).

Matched Pairs

Data: Consider pre- and post-test scores for 14 students enrolled in a course.

pre=c(52,49,55,37,68,72,64,71,81,57,52,66,71,65)
post=c(56,61,65,43,65,78,77,73,82,61,54,63,71,72)

Are students improving test scores as a result of the course?

Let \(\mu_{\text{diff}}=\) the average difference (post-pre) in test scores for all students who take the course.

Note: The pre and post data need to have been ordered to match according to student for this to work. So, the first student had pre and post-test scores of 52 and 56, the second student had pre and post-test scores of 49 and 61, etc.

We test \[H_o: \mu_{\text{diff}}=0~~\text{against}~~ H_a: \mu_{\text{diff}}>0,\] by doing a 1-sample \(t\)-test on the difference

diff=post-pre
t.test(diff,mu=0,alternative="greater")
## 
##  One Sample t-test
## 
## data:  diff
## t = 3.2495, df = 13, p-value = 0.003167
## alternative hypothesis: true mean is greater than 0
## 95 percent confidence interval:
##  1.982586      Inf
## sample estimates:
## mean of x 
##  4.357143

Conclusion: With such a small p-value we reject the null hypothesis in favor of the alternative and conclude that the average difference (post-test minus pre-test) is positive, suggesting the course helps improve test scores, on average!

Two means

Question: Is there a difference between two population means?

Example: Is there a difference in the silver content (% Ag) in coins during two different eras of the reign of King Manuel, I, Comnenus (1143-1180)? 16 coins have been found and measured, 9 of the coins came from an early coinage, and 7 of the coins came from a coinage many years later.

Byzantine coin
Byzantine coin

Parameters of Interest

  • \(\mu_1\) - the mean silver content of all coins produced in the early coinage
  • \(\mu_2\) - the mean silver content of all coins produced in the later coinage
  • In fact, we’ll be interested in the difference \(\mu_1 - \mu_2\).

Estimating \(\mu_1-\mu_2\) from a simple random sample

From 2 independent simple random samples drawn from the two populations we find these six values:

  • \(n_1\), \(\overline{x}_1\), \(s_1\) - the sample size, sample mean, and sample standard deviation of the sample gathered from the first population.
  • \(n_2\), \(\overline{x}_2\), \(s_2\) - the sample size, sample mean, and sample standard deviation of the sample gathered from the first population.

The Data

Ag1 <- c(5.9,6.8,6.4,7.0,6.6,7.7,7.2,6.9,6.2)
Ag2 <- c(5.3,5.6,5.5,5.1,6.2,5.8,5.8)

These data appear in The Handbook of Small Data Sets (p. 118), and are based on this article:

Hendy, M.F. and Charles J.A. (1970), The production techniques, silver content and circulation history of the twelfth-century Byzantine Trachy. Archaeonetry, 12. 13-21)

Summary Statistics

#first sample
n1 = length(Ag1)
xbar1 = mean(Ag1)
s1 = sd(Ag1)

#second sample
n2 = length(Ag2)
xbar2 = mean(Ag2)
s2 = sd(Ag2)

Confidence Interval for \(\mu_1-\mu_2\)

We use the formula \[(\overline{x}_1-\overline{x}_2) \pm t^*\cdot \text{SE},\] where \[\text{SE}=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\]

Calculate the confidence interval in R

#first sample
n1 = length(Ag1)
xbar1 = mean(Ag1)
s1 = sd(Ag1)

#second sample
n2 = length(Ag2)
xbar2 = mean(Ag2)
s2 = sd(Ag2)

# Confidence Level
C=.95
# Finding t* with rule of thumb
df = min(n1-1,n2-1)
tstar=qt(C+(1-C)/2,df)
SE = sqrt(s1^2/n1 + s2^2/n2) # standard error
moe=tstar*SE # margin of error
(xbar1-xbar2)+c(-moe,moe) #confidence interval
## [1] 0.5744126 1.6859049

Interpretation We believe with 95% confidence that the difference in the average silver content from the early coinage and the average silver content from the later coinage is between 0.57 and 1.69. Since this entire interval consists of positive values, we believe the average silver content in coins produced early in the reign is greater than the average silver content in coins produces later in the reign.

Significance Test

To test \[H_o: \mu_1 - \mu_2 = 0 ~~\text{against}~~ H_a: \mu_1 - \mu_2 \neq 0,\] the 2-sample test statistic is given by the formula \[t = \frac{\overline{x}_1-\overline{x}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\]

Calculate the test statistic in R

#first sample
n1 = length(Ag1)
xbar1 = mean(Ag1)
s1 = sd(Ag1)

#second sample
n2 = length(Ag2)
xbar2 = mean(Ag2)
s2 = sd(Ag2)

## Test statistic
SE = sqrt(s1^2/n1 + s2^2/n2) # standard error
t = (xbar1-xbar2)/SE
t
## [1] 4.976011

t.test()

t.test(Ag1,Ag2,  alternative="two.sided")
## 
##  Welch Two Sample t-test
## 
## data:  Ag1 and Ag2
## t = 4.976, df = 13.765, p-value = 0.0002138
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.6422519 1.6180655
## sample estimates:
## mean of x mean of y 
##  6.744444  5.614286

Conclusion With such a small p-value we reject the null hypothesis in favor of the alternative and conclude that there is a difference in average silver content of coins produced during these two eras.

Note on df: For the 2-sample t-test, R’s built-in t.test() function results is the same test statistic as the one we produce following the formula in the book, but it assumes a different, less conservative degrees of freedom than what we’ve done in class This creates a slightly different, less conservative margin of error in confidence intervals, and a slightly different, less conservative p-value in significance tests, than what we do by hand. If interested, see the course notes for the df formula that R uses.

t.test() and data frames

Data from on-line surveys such as you’d build with Google Forms is stored as a long data frame, where each row in the spreadsheet contains all the answers from an individual respondent, and each column has all the responses to a particular question. In long form, the coin data in the example above would look like this:

coinage Ag
early 5.9
early 6.8
early 6.4
early 7.0
early 6.6
early 7.7
early 7.2
early 6.9
early 6.2
late 5.3
late 5.6
late 5.5
late 5.1
late 6.2
late 5.8
late 5.8

We can use the t.test() command as follows to run the 2-sample \(t\)-test when the data are given as a long form data matrix:

t.test(Ag~coinage,data=df,alternative="two.sided")
## 
##  Welch Two Sample t-test
## 
## data:  Ag by coinage
## t = 4.976, df = 13.765, p-value = 0.0002138
## alternative hypothesis: true difference in means between group early and group late is not equal to 0
## 95 percent confidence interval:
##  0.6422519 1.6180655
## sample estimates:
## mean in group early  mean in group late 
##            6.744444            5.614286

Also, if we load the tidyverse (with library(tidyverse)) into our session, we can easily extract sample sizes and sample means and sample standard deviations by group as follows:

df %>% 
  group_by(coinage) %>%
  summarize(size=length(Ag),avg=mean(Ag),stdev=sd(Ag)) 
## # A tibble: 2 × 4
##   coinage  size   avg stdev
##   <chr>   <int> <dbl> <dbl>
## 1 early       9  6.74 0.543
## 2 late        7  5.61 0.363

Two proportions

Question: We want to estimate/investigate the difference between two population proportions \(p_1 - p_2\).

Example: Is there a difference between the proportion of all Linfield Students that have tested positive for COVID-19 at least once and the proportion of all Oregonians that have tested positive for COVID-19 at least once?

Parameters of Interest

  • \(p_1\) - proportion of all elements in population 1 having the feature under investigation.
  • \(p_2\) - the proportion of all elements in population 2 having the feature under investigation.
  • In fact, we’ll be interested in the difference \(p_1 - p_2\).

Estimating \(p_1-p_2\) from a simple random sample

From 2 independent simple random samples drawn from the two populations we find these values:

  • \(n_1\), and \(\hat{p}_1\) - the sample size and sample proportion from population 1 that have the feature.
  • \(n_2\) and \(\hat{p_2}\) - the sample size and sample proportion from population 2 that have the feature.

Confidence Interval for \(p_1-p_2\)

The formula for the confidence interval:

\[(\hat{p_1}-\hat{p_2}) \pm z^* \cdot \text{SE}\] where \[SE \approx \sqrt{\frac{\hat{p_1}(1-\hat{p_1})}{n}+\frac{\hat{p_2}(1-\hat{p_2})}{n}}.\]

Calculate the interval in R

# (using artificial data)
n1=64 #sample size from pop'n 1
x1=29
phat1=x1/n1 
n2=150 #sample size from pop'n 2
x2=42
phat2=x2/n2
C = .95 # confidence level
SE = sqrt(phat1*(1-phat1)/n1+phat2*(1-phat2)/n2)
zstar = qnorm(C + (1-C)/2)
MOE = zstar*SE #margin of error
round((phat1-phat2) + c(-MOE,MOE),3) #Confidence Interval
## [1] 0.032 0.315

Interpretation: We believe that the difference in population proportions, \(p_1 - p_2\), is in the range provided by the interval. Since the entire range of values in the interval is positive, the interval provides strong evidence that the difference \(p_1-p_2\) is positive. In other words, we have reason to believe that \(p_1 > p_2\). We have reason to believe that the proportion of Linfield students that have tested positive for COVID-19 at least once is greater than the state-wide proportion.

Hypothesis test

To test the null hypothesis \[H_o: p_1-p_2 = 0\] against one of these three alternative hypotheses:

  • \(H_a: p_1-p_2 > 0\) (one-sided, greater than \(p_o\))
  • \(H_a: p_1-p_2 < 0\) (one-sided, less than \(p_o\))
  • \(H_a: p_1-p_2 \neq 0\) (two.sided)

Test statistic: This is a z-score computed under the assumption that \(H_o\) is true: \[z = \frac{\hat{p_1}-\hat{p_2}}{\text{SE}}\] where \[SE \approx \sqrt{\frac{\hat{p}(1-\hat{p})}{n}+\frac{\hat{p}(1-\hat{p})}{n}},\] and in this formula, \(\hat{p}\) denotes the pooled sample proportion, \(\displaystyle\hat{p}=\frac{x_1+x_2}{n_1+n_2}\).

Suppose we want to test the claim that the proportion of Linfield students who have tested positive for COVID-19 is the same as the state-wide proportion. We choose the hypotheses \[H_o: p_1-p_2 = 0 ~~\text{against}~~H_a: p_1-p_2 \neq 0.\]

Calculate the test statistic in R

# population 1
n1=64 
x1=29 
phat1 = x1/n1
#population 2
n2=150 
x2=42
phat2=x2/n2
#pooled sample proportion
phat=(x1+x2)/(n1+n2)
C = .95 # confidence level
SE = sqrt(phat*(1-phat)/n1+phat*(1-phat)/n2)
z = (phat1-phat2)/SE
z
## [1] 2.462664

We can determine the p-value with the pnorm() function. How to use the pnorm() function depends on the alternative hypothesis.

Chi-square Goodness of Fit

Question: Do observations in a random sample follow a known distribution?

Example: Consider the population of Oregon towns and cities, and focus on the leading digit of each of these population numbers, as in the table below. Note: Data found here.

We display the first four and last four rows of this large data frame (that has 373 rows) The data frame is called orcities.
city pop_year population lead_digit
Portland 2023 630498 6
Eugene 2023 177899 1
Salem 2023 177432 1
Gresham 2023 110685 1
Hillsboro 2023 107730 1
Bend 2023 104557 1
Beaverton 2023 96945 9
Medford 2023 85098 8
Corvallis 2023 61087 6
Springfield 2023 61085 6

We ask this question of the data: Is it reasonable to expect that each of the digits 1-9 occur with equal frequency? A chi-square goodness of fit test can answer this question.

Below is the table of observed first digit frequencies, which we can obtain from the data frame above by running

table(orcities$lead_digit)
## 
##   1   2   3   4   5   6   7   8   9 
## 150  72  55  34  28  28  14  20  18

Run the chi-square test in R

X=chisq.test(table(orcities$lead_digit),p = c(rep(1/9,9)))

Expected counts (rounded to 1 decimal point):

round(X$expected,1)
##    1    2    3    4    5    6    7    8    9 
## 46.6 46.6 46.6 46.6 46.6 46.6 46.6 46.6 46.6

If all nine digits 1-9 arise with equal probability (1/9), we would expect about 41.4 occurrences of each digit in a sample of size 373. The observed counts appear to have a much different distribution than this.

Results:

X
## 
##  Chi-squared test for given probabilities
## 
## data:  table(orcities$lead_digit)
## X-squared = 318.89, df = 8, p-value < 2.2e-16

Conclusion: With such a small p-value (essentially 0, and well, well below \(\alpha = .05\))) we reject the claim that all 9 digits 1-9 appear with equal frequency in a distribution of city populations.

Example 2. Benford’s Law asserts that in many distributions (but not all) the distribution of “leading digits” follows this specific, non-uniform distribution:

Leading Digit 1 2 3 4 5 6 7 8 9
probability .301 .176 .125 .097 .079 .067 .058 .051 .046

(These probabilities might seem artifical, but, according to Benford’s Law, the probability that a leading digit is \(k\) is given mathematically by \(\log_{10}\left(\frac{k+1}{k}\right)\).)

Is it plausible that Oregon town populations adhere to this distribution?

X=chisq.test(table(orcities$lead_digit),p = c(.301,.176,.125,.097,.079,.067,.058,.051,.046))

Expected counts:

X$expected
##       1       2       3       4       5       6       7       8       9 
## 126.119  73.744  52.375  40.643  33.101  28.073  24.302  21.369  19.274

These expected counts look much closer to the observed counts! Let’s view the results of the test:

Here are the results of the chi-square test:

X
## 
##  Chi-squared test for given probabilities
## 
## data:  table(orcities$lead_digit)
## X-squared = 11.106, df = 8, p-value = 0.1958

Conclusion: Here the p-value is larger than .05, suggesting we do not have strong evidence to reject the null hypothesis that population figures in Oregon follow Benford’s Law. Perhaps Benford’s Law is a reasonable model for the leading digit counts for Oregon city and town populations.

Chi-square Test of Independence

Question: Are two categorical variables associated?

Example: Is there an association between the wall color in a room and the mood of people resting in the room?

The Data

Here are the first five rows of the data frame, called df, which has 100 rows total.

color mood
white anxious
red calm
yellow happy
white anxious
white anxious

Here’s the two-way table:

table(df$mood,df$color)
##          
##           purple red white yellow
##   anxious     14   8    12      5
##   calm        10   7    10      7
##   happy        8   7     4      8

Conducting the chi-square test of independence:

X = chisq.test(table(df$mood,df$color))
# expected cell counts under the assumption that there is no association
X$expected
##          
##           purple  red white yellow
##   anxious  12.48 8.58 10.14    7.8
##   calm     10.88 7.48  8.84    6.8
##   happy     8.64 5.94  7.02    5.4

The \(\chi^2\) test statistic

X$statistic
## X-squared 
##  4.618345

The summary of the test:

X
## 
##  Pearson's Chi-squared test
## 
## data:  table(df$mood, df$color)
## X-squared = 4.6183, df = 6, p-value = 0.5936

Conclusion: We have no statistical evidence of an association (since the p-value is so large, \(P>.5\)).

ANOVA

Question: We are interested in comparing more than two population means.

Example: We investigate whether there is a difference in average weight-loss for participants in different exercise programs: one that focuses on long exercise periods, one focusing on short exercise periods, and one focusing on short exercise periods with equipment.

Notation/terminology

  • \(k\) denotes the number of groups we’re comparing
  • \(\mu_i\) - the (unknown) population mean for group \(i\) (where \(i = 1, 2, \ldots, k\))
  • \(n_i\) - the sample size of the independent sample gathered from group \(i\).
  • \(\overline{x}_i\) - the sample mean of the \(n_i\) values in the sample from group \(i\).
  • \(s_i\) - the sample standard deviation of the \(n_i\) values in the sample from group \(i\).
  • \(N\) the overall sample size (\(N = n_1 + n_2 + \cdots + n_k\))
  • MSG - the “mean square for groups” which measures the variation across all the group sample means
  • MSE = the “mean square for error” which measures the overall variation of all \(N\) data points

The Data

It is convenient to encode the data as a long form data matrix, in which each row corresponds to a participant in the study, and we have two columns: one that specifies which group the participant was in, and one that records the value of interest (weight-loss in our example). Here are the first ten rows of the data frame:

program weight.loss
Equip 6.0
Long 3.1
Short 10.4
Short 11.1
Equip 12.0
Equip 3.0
Equip 8.2
Equip 10.9
Equip 8.7
Equip 12.8

With the tidyverse loaded into your session, here is a quick way to extract summary statistics (the data frame with the program and weight.loss columns is called df)

df %>% 
  group_by(program) %>%
  summarize(size=length(weight.loss),
            avg=mean(weight.loss),
            stdev=sd(weight.loss))
## # A tibble: 3 × 4
##   program  size   avg stdev
##   <chr>   <int> <dbl> <dbl>
## 1 Equip      42  9.26  4.33
## 2 Long       37 11.0   4.34
## 3 Short      36  8.48  3.12

We note that the number of groups in this study is \(k = 3\), and the total sample size is \(N = 115\).

Side-by-side box plots:

ggplot(df)+
  geom_boxplot(aes(x=program,y=weight.loss,fill=program),show.legend=FALSE)+
  ggtitle("Weight loss by exercise program")

Running anova in R:

anova(lm(weight.loss~program,data=df))
## Analysis of Variance Table
## 
## Response: weight.loss
##            Df  Sum Sq Mean Sq F value  Pr(>F)  
## program     2  119.26  59.632    3.74 0.02677 *
## Residuals 112 1785.78  15.944                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Conclusion: From the print out, the test statistic from our data is \(F = 3.74\), and this statistic lives in an F-distribution with numerator df = 2 and denominator df = 112.
The p-value for this test is .02677, which indicates the probability of having observed such a difference among group sample means if in fact all groups have the same population mean. Since this likelihood is so small (indeed it’s less than .05), we have reason to reject the null and conclude that there is a difference in average weight-loss for these different exercise programs. We note that the components of \(F = MSG/MSE\) appear in the display above. MSG = 59.632 and MSE = 15.944.