Functions

NMstats is an R package that contains functions aligned to the Navidi/Monk Elementary & Essential Statistics textbooks. This vignette describes the functions available along with special considerations and examples.


Call the NMstats package:

library(NMstats)

combs

The combs function calculates the number of ways to choose r items from a total of n without replacement and where order does not matter.

 

Usage

The format is combs(n, r) where n is the total number of items and r is the number of items to choose.

 

Return

The function returns an integer value that represents the number of combinations of r items chosen from n. If invalid input values are given, an error message is returned.

 

Examples

Count the number of combinations of 4 items chosen from 10 

combs(10, 4)
## [1] 210

Computation of binomial probability where n = 15, p = 0.25, and x = 6. 

n <- 15
p <- 0.25
x <- 6
prob_value <- combs(n, x) * p^x * (1 - p)^(n - x)
print(prob_value)
## [1] 0.09174777




perms

The perms function calculates the number of ways to choose r items from a total of n without replacement and where order matter.

 

Usage

The format is perms(n, r) where n is the total number of items and r is the number of items to choose.

 

Return

The function returns an integer value that represents the number of permutations of r items chosen from n. If invalid input values are given, an error message is returned.

 

Examples

The number of permutations of 7 items chosen from 12. 

perms(12, 7)
## [1] 3991680

Computation of the probability of guessing a 3-digit number (allowing zeros) if none of the digits are repeated. 

num_outcomes <- perms(10, 3)
prob_value <- 1/num_outcomes
print(prob_value)
## [1] 0.001388889




outlier_bounds

The outlier_bounds function computes the lower and upper outlier boundaries using the IQR method.

 

Usage

The format is outlier_bounds(data) where data is a data set.

 

Return

The function will print the lower and upper outlier bounds. Variables Lower.bound and Upper.bound are available as return values after execution of this function.

 

Examples

Print the lower and upper outlier bounds of a data set 

data <- c(14, 9, 3, 22, 8, 13, 6)
outlier_bounds(data)
## 
## Lower Outlier Bound: -2.75 
## Upper Outlier Bound: 23.25

Calling lower and upper outlier bounds after result <- outlier_bounds(data) 

result$Lower.bound
## [1] -2.75
result$Upper.bound
## [1] 23.25




data_range

The data_range function computes the range of a data set.

 

Usage

The format is data_range(data) where data is a data set.

 

Return

The function returns the range of a data set, defined as the minimum subtracted from the maximum.

 

Examples

Return the range of a data set 

data <- c(24, -67, 15, 89, -34, 51, -42, 76)
data_range(data)
## [1] 156

Comparing the ranges of two data sets 

data_1 <- c(12.4, -5.3, 8.7, 19.1, 2.9)
data_2 <- c(-23.1, 7.5, 14.2, 3.8, -6.7)
data_range(data_1)
## [1] 24.4
data_range(data_2)
## [1] 37.3




rel_hist

The rel_hist function constructs a relative frequency histogram for a data set.

 

Usage

The format is rel_hist(data, bins, col, xlab, ylab, main, ybreaks) where data is a data set.

Optional arguments are:

  • bins is the number of bins to use
  • col is the fill color
  • xlab and ylab are the labels for the axes
  • main is the title of the graph
  • ybreaks is a numeric vector specifying the tick marks on the y-axis.

 

Return

The function constructs a relative frequency histogram.

 

Examples

Generate random data and construct relative histogram 

data <- sample(1:100, 220, replace = TRUE)
rel_hist(data, bins = 20, xlab = "My Data")




var.p

The var.p function calculates the population variance of a data set.

 

Usage

The format is var.p(data) where data is a data set.

 

Examples

Calculate the population variance of a data set. 

data <- c(37, 292, 175, 86, 331, 249, 104, 58, 368, 213)
var.p(data)
## [1] 12523.21




sd.p

The sd.p function calculates the population standard deviation of a data set.

 

Usage

The format is sd.p(data) where data is a data set.

 

Examples

Calculate the population standard deviation of a data set. 

data <- c(-3.2, 7.1, -12.5, 4.8, -0.6, 11.9, -9.4, 2.3, 6.7)
sd.p(data)
## [1] 7.563427

Computation of a z-score of a data value from a population 

data <- c(3.1, 5.2, 6.3, 4.4, 2.8, 7.9, 1, 8.6, 0.7, 9.5, 4.1, 4)

x_val <- 2.8
# z-score
(x_val - mean(data))/sd.p(data)
## [1] -0.7386329




Z_Interval

The Z_Interval function calculates the confidence interval for a population mean when the population standard deviation is known.

 

Usage

The format is Z_Interval(xbar, n, sigma, alpha) where xbar is the sample mean, n is the sample size, sigma is the population standard deviation, and alpha is a value between 0 and 1 where 1 - alpha is the confidence level as a decimal.

For example, if alpha = 0.02, this represents a 98% confidence level. If alpha is omitted, a default value of 0.05 is used, representing a 95% confidence level for the interval.

 

Return

The function will print the confidence level, margin or error, critical value, and the confidence interval (lower and upper bounds).

Variables critval, Lbound, and Ubound are available as return values after execution of this function.

If invalid input values are given, an error message is returned.

 

Examples

Construct a 90% confidence interval of a sample with size \(n = 36\) and mean \(\bar{x} = 39.8\) from a population with known standard deviation \(\sigma =6.4\). 

Z_Interval(xbar = 39.8, n = 36, sigma = 6.4, alpha = 0.1)
## Confidence Level: 90 % 
## Margin of Error: 1.75451 
## Critical Value: 1.64485 
## Lower Bound: 38.04549 
## Upper Bound: 41.55451

Construct a 98% confidence interval of a sample that comes from a population with standard deviation \(\sigma = 12.8\). 

# Define sample data
sample_data <- c(17.4, 28.9, 52.9, 29.3, 21.1, 16.5, 16.9, 16.2, 15.7, 14.2,
    26.7, 38.6, 23.6, 24.3, 22.9, 25.9, 24.4, 28.3, 4.9, 27.5)

# Construct confidence interval
Z_Interval(xbar = mean(sample_data), n = length(sample_data), sigma = 12.8,
    alpha = 0.02)
## Confidence Level: 98 % 
## Margin of Error: 6.6584 
## Critical Value: 2.32635 
## Lower Bound: 17.1516 
## Upper Bound: 30.4684




T_Interval

The T_Interval function calculates the confidence interval for a population mean when the population standard deviation is not known, and the sample standard deviation is used instead. The confidence interval is based on the Student’s t-distribution.

 

Usage

The format is T_Interval(xbar, n, s, alpha) where xbar is the sample mean, n is the sample size, s is the sample standard deviation, and alpha is a value between 0 and 1 where 1 - alpha is the confidence level as a decimal.

For example, if alpha = 0.02, this represents a 98% confidence level. If alpha is omitted, a default value of 0.05 is used, representing a 95% confidence level for the interval.

 

Return

The function will print the confidence level, margin or error, number of degrees of freedom, critical value, and the confidence interval (lower and upper bounds).

Variables critval, Lbound, and Ubound are available as return values after execution of this function.

If invalid input values are given, an error message is returned.

 

Examples

Construct a 95% confidence interval of a sample with size \(n = 129\), and mean \(\bar{x} = 67.6\), and sample standard deviation \(s =18.2\). 

T_Interval(xbar = 67.6, n = 129, s = 18.2, alpha = 0.05)
## Confidence Level: 95 % 
## Margin of Error: 3.17066 
## Number of Degrees of Freedom: 128 
## Critical Value: 1.97867 
## Lower Bound: 64.42934 
## Upper Bound: 70.77066

Construct a 90% confidence interval of a sample. 

# Define sample data
exam_grades <- c(64, 70, 67, 60, 74, 63, 73, 72, 66, 71, 60, 65, 65, 76,
    56, 70, 68, 67, 67, 62, 63, 60, 79, 69, 75)

# Construct confidence interval
T_Interval(xbar = mean(exam_grades), n = length(exam_grades), s = sd(exam_grades),
    alpha = 0.1)
## Confidence Level: 90 % 
## Margin of Error: 1.9495 
## Number of Degrees of Freedom: 24 
## Critical Value: 1.71088 
## Lower Bound: 65.3305 
## Upper Bound: 69.2295




One_Prop_Int

The One_Prop_Int function calculates the confidence interval for a population proportion. The confidence interval is based on the normal distribution.

 

Usage

The format is One_Prop_Int(x, n, alpha) where x is the number of individuals of interest in the sample, n is the sample size, and alpha is a value between 0 and 1 where 1 - alpha is the confidence level as a decimal.

For example, if alpha = 0.02, this represents a 98% confidence level. If alpha is omitted, a default value of 0.05 is used, representing a 95% confidence level for the interval.

 

Return

The function will print the confidence level, margin or error, critical value, sample proportion, and the confidence interval (lower and upper bounds).

Variables sprop, critval, Lbound, and Ubound are available as return values after execution of this function.

If invalid input values are given, an error message is returned.

 

Examples

Construct a 92% confidence interval when \(x = 44\) and \(n = 192\). 

One_Prop_Int(x = 44, n = 192, alpha = 0.08)
## Confidence Level: 92 % 
## Margin of Error: 0.0531 
## Critical Value: 1.75069 
## Sample Proportion: 0.22917 
## Lower Bound: 0.17606 
## Upper Bound: 0.28227

Construct a 90% confidence interval for the population proportion of the number of “Yes” in a sample. 

# Define sample data
sample_data <- c("No", "No", "No", "Yes", "Yes", "Yes", "No", "Yes", "No",
    "No", "No", "No", "No", "Yes", "No", "No", "No", "Yes", "No", "Yes",
    "No", "Yes", "Yes", "No", "No", "Yes", "Yes", "No", "No", "Yes", "Yes",
    "No", "No", "No", "Yes", "No", "No", "Yes", "Yes", "Yes", "Yes", "Yes",
    "Yes", "Yes", "No", "No", "Yes", "No", "Yes", "Yes", "Yes", "No", "No",
    "Yes", "No", "No", "No", "Yes", "No", "Yes")

# Summarize data into a table
table(sample_data)
## sample_data
##  No Yes 
##  32  28

# Extract yes's, unname, and assign to x
x <- unname(table(sample_data)["Yes"])

# Determine sample size
n <- length(sample_data)

# Explicitly state x (number of individuals of interest) and n (sample
# size)
x
## [1] 28
n
## [1] 60

# Construct 90% confidence interval
One_Prop_Int(x, n, alpha = 0.1)
## Confidence Level: 90 % 
## Margin of Error: 0.10594 
## Critical Value: 1.64485 
## Sample Proportion: 0.46667 
## Lower Bound: 0.36073 
## Upper Bound: 0.57261




Z_Test

The Z_Test function performs a hypothesis test about a population mean when the population standard deviation is known. P-values are calculated based on the normal distribution.

 

Usage

The format is Z_Test(xbar, n, sigma, mu, alt) where xbar is the sample mean, n is the sample size, sigma is the known population standard deviation, mu is the hypothesized mean in the null hypothesis, and alt is the form of the alternate hypothesis.

alt is a character string with choices “left”, “right”, or “two”. The choices “less”, “greater”, or “two.sided” are also accepted.

 

Return

The function will print the test statistic z and the p-value.

Variables z and pvalue are available as return values after execution of this function.

If invalid input values are given, an error message is returned.

 

Examples

Perform a hypothesis test of \(H_0: \mu = 50\) versus \(H_1: \mu < 50\) where \(\bar{x} = 47.3\), \(n = 44\), and \(\sigma = 10.2\). 

Z_Test(xbar = 47.3, n = 44, sigma = 10.2, mu = 50, alt = "left")
## Test Statistic: z = -1.75586 
## P-Value: 0.03956

Perform a hypothesis test of \(H_0: \mu = 67.9\) versus \(H_1: \mu \ne 67.9\) given a sample data set from a population with known standard deviation \(\sigma = 14\). 

# Define sample data
my_data <- c(73, 92, 55, 85, 59, 87, 77, 52, 67, 88, 89, 54, 75, 68, 80,
    90, 66, 78, 55, 86, 82, 63, 70, 84, 58, 53, 60, 91, 57, 71, 81, 69)

# Perform hypothesis test
Z_Test(xbar = mean(my_data), n = length(my_data), sigma = 14, mu = 67.9,
    alt = "two")
## Test Statistic: z = 1.79555 
## P-Value: 0.07257




T_Test

The T_Test function performs a hypothesis test about a population mean when the population standard deviation is not known, and the sample standard deviation is used instead. P-values are calculated based on the Student’s t-distribution.

 

Usage

The format is T_Test(xbar, n, s, mu, alt) where xbar is the sample mean, n is the sample size, s is the sample standard deviation, mu is the hypothesized mean in the null hypothesis, and alt is the form of the alternate hypothesis.

alt is a character string with choices “left”, “right”, or “two”. The choices “less”, “greater”, or “two.sided” are also accepted.

 

Return

The function will print the number of degrees of freedom, the test statistic t, and the p-value.

Variables t and pvalue are available as return values after execution of this function.

If invalid input values are given, an error message is returned.

 

Examples

Perform a hypothesis test of \(H_0: \mu = 192\) versus \(H_1: \mu > 192\) where \(\bar{x} = 199.7\), \(n = 60\), and \(s = 41.8\). 

T_Test(xbar = 199.7, n = 60, s = 41.8, mu = 192, alt = "right")
## Number of Degrees of Freedom: 59 
## Test Statistic: t = 1.42689 
## P-Value: 0.07944

Perform a hypothesis test of \(H_0: \mu = 71\) versus \(H_1: \mu \ne 71\) given a sample data set (assume population is normally distributed). 

# Define sample data
my_data <- c(66, 71, 100, 76, 77, 102, 82, 55, 64, 68, 95, 81, 81, 77, 66,
    104, 83, 44, 86, 67, 58, 72, 59, 63, 65, 48, 88, 77, 57, 95, 82, 70,
    89, 89, 88, 86, 84, 74, 70, 69)


# Perform hypothesis test
T_Test(xbar = mean(my_data), n = length(my_data), s = sd(my_data), mu = 71,
    alt = "two")
## Number of Degrees of Freedom: 39 
## Test Statistic: t = 2.07723 
## P-Value: 0.04441




One_Prop_Test

The One_Prop_Test function performs a one-sample hypothesis test about a population proportion.

 

Usage

The format is One_Prop_Test(x, n, p0, alt) where x is the number of individuals of interest in the sample, n is the sample size, p0 is the hypothesized proportion in the null hypothesis, and alt is the form of the alternate hypothesis.

alt is a character string with choices “left”, “right”, or “two”. The choices “less”, “greater”, or “two.sided” are also accepted.

 

Return

The function will print the sample proportion, the test statistic z, and the p-value.

Variables sprop, z and pvalue are available as return values after execution of this function.

If invalid input values are given, an error message is returned.

 

Examples

Perform a hypothesis test of \(H_0: p = 0.5\) versus \(H_1: p < 0.5\) where \(x = 480\) and \(n = 1000\). 

One_Prop_Test(x = 480, n = 1000, p0 = 0.5, alt = "left")
## Sample Proportion: 0.48 
## Test Statistic: z = -1.26491 
## P-Value: 0.10295

Perform a hypothesis test of \(H_0: p = 0.45\) versus \(H_1: p > 0.45\) for the population proportion of the number of “B” in a sample. 

# Define sample data
samp <- c("A", "B", "B", "B", "B", "A", "A", "A", "A", "A", "B", "B", "A",
    "B", "B", "B", "A", "A", "B", "B", "A", "A", "A", "B", "A", "B", "B",
    "A", "A", "B", "B", "B", "A", "B", "A", "B", "B", "A", "B", "A", "B",
    "B", "B", "A", "B", "B", "A", "B", "A", "B", "B", "B", "B", "A", "A",
    "A", "B", "B", "B", "B")


# Summarize data into a table
table(samp)
## samp
##  A  B 
## 25 35

# Extract B's, unname, and assign to x
x <- unname(table(samp)["B"])

# Determine sample size
n <- length(samp)

# Explicitly state x (number of individuals of interest) and n (sample
# size)
x
## [1] 35
n
## [1] 60

# Perform the hypothesis test
One_Prop_Test(x, n, p0 = 0.45, alt = "right")
## Sample Proportion: 0.58333 
## Test Statistic: z = 2.076 
## P-Value: 0.01895




Two_Samp_T_Interval

The Two_Samp_T_Interval function constructs a confidence interval for the difference between two population means (mu_1 - mu_2) given two independent samples. This confidence interval is based on the Student’s t-distribution.

 

Usage

The format is Two_Samp_T_Interval(xbar1, s1, n1, xbar2, s2, n2, alpha) where xbar1 is the mean of Sample 1, n1 is the size of Sample 1, s1 is the standard deviation of Sample 1, xbar2 is the mean of Sample 2, n2 is the size of Sample 2, s2 is the standard deviation of Sample 2, and alpha is a value between 0 and 1 where 1 - alpha is the confidence level as a decimal.

 

Return

The function will print the confidence level, margin of error, number of degrees of freedom, critical value, point estimate, and the confidence interval (lower and upper bounds).

Variables critval, point_est, Lbound, and Ubound are available as return values after execution of this function.

If invalid input values are given, an error message is returned.

 

Examples

Construct a 99% confidence interval for \(\mu_1 - \mu_2\) where \(\bar{x}_1 = 10\), \(s_1 = 2.5\), \(n_1 = 125\), \(\bar{x}_2 = 8.6\), \(s_2 = 3.4\), \(n_2 = 131\) and the samples are independent. 

Two_Samp_T_Interval(xbar1 = 10, s1 = 2.5, n1 = 125, xbar2 = 8.6, s2 = 3.4,
    n2 = 131, alpha = 0.01)
## Confidence Level: 99 % 
## Margin of Error: 0.96544 
## Number of Degrees of Freedom: 238.7094 
## Critical Value: 2.59658 
## Point Estimate: 1.4 
## Lower Bound: 0.43456 
## Upper Bound: 2.36544

Construct a 90% confidence interval for \(\mu_1 - \mu_2\) given two independent samples. 

# Define Sample 1
exam_1 <- c(58, 67, 61, 62, 78, 47, 59, 72, 89, 68, 37, 81, 64, 64, 62, 70,
    80, 67, 48, 68, 81, 84, 62, 76, 52, 92, 72, 62, 79, 61, 56, 61, 67, 76,
    61, 65, 77, 62, 72, 71, 57, 55, 58, 44, 62, 83, 73, 55, 60, 74)

# Define Sample 2
exam_2 <- c(76, 47, 31, 35, 69, 42, 79, 60, 56, 74, 25, 69, 79, 37, 48, 58,
    30, 54, 43, 57, 48, 62, 49, 44, 57, 61, 61, 56, 41, 31, 78, 59, 35, 31,
    47, 47, 53, 69, 28, 76, 59, 51, 57, 38, 70, 64)

# Calculate sample statistics
xbar1 <- mean(exam_1)
n1 <- length(exam_1)
s1 <- sd(exam_1)
xbar2 <- mean(exam_2)
n2 <- length(exam_2)
s2 <- sd(exam_2)

# Construct confidence interval
Two_Samp_T_Interval(xbar1, s1, n1, xbar2, s2, n2, alpha = 0.1)
## Confidence Level: 90 % 
## Margin of Error: 4.57168 
## Number of Degrees of Freedom: 83.75668 
## Critical Value: 1.66325 
## Point Estimate: 13.17478 
## Lower Bound: 8.6031 
## Upper Bound: 17.74646




Two_Samp_T_Test

The Two_Samp_T_Test function performs a hypothesis test about the difference between two population means (mu_1 - mu_2) given two independent samples. P-values are calculated based on the Student’s t-distribution.

 

Usage

The format is Two_Samp_T_Test(xbar1, s1, n1, xbar2, s2, n2, alt, mu) where xbar1 is the mean of Sample 1, n1 is the size of Sample 1, s1 is the standard deviation of Sample 1, xbar2 is the mean of Sample 2, n2 is the size of Sample 2, s2 is the standard deviation of Sample 2, mu is the hypothesized difference in the null hypothesis, and alt is the form of the alternate hypothesis.

alt is a character string with choices “left”, “right”, or “two”. The choices “less”, “greater”, or “two.sided” are also accepted.

 

Return

The function will print the number of degrees of freedom, the test statistic t, and the p-value.

Variables t and pvalueare available as return values after execution of this function.

If invalid input values are given, an error message is returned.

 

Examples

Perform a hypothesis test for \(H_0: \mu_1 = \mu_2\) versus \(H_1: \mu_1 \ne \mu_2\) where \(\bar{x}_1 = 34.8\), \(s_1 = 5.2\), \(n_1 = 55\), \(\bar{x}_2 = 32.8\), \(s_2 = 3.8\), \(n_2 = 61\) and the samples are independent. 

Two_Samp_T_Test(xbar1 = 34.8, s1 = 5.2, n1 = 55, xbar2 = 32.8, s2 = 3.8,
    n2 = 61, alt = "two", mu = 0)
## Number of Degrees of Freedom: 98.06019 
## Test Statistic: t = 2.34346 
## P-Value: 0.02112

Perform a hypothesis test for \(H_0: \mu_1 = \mu_2\) versus \(H_1: \mu_1 < \mu_2\) given two independent samples. 

# Define Sample 1
old_process <- c(1075, 1012, 980, 1035, 1001, 978, 965, 1030, 990, 1023,
    1015, 998, 1075, 1008, 1085, 976, 1105, 1002, 987, 1021, 992, 1010, 1003,
    1006, 1090, 988, 1001, 965, 1007, 973, 998, 982, 965, 1015, 987, 1000,
    1089, 1109, 995, 1104, 1075, 1018, 1025, 981, 1077, 995)

# Define Sample 2
new_process <- c(1048, 966, 1029, 1098, 1064, 1035, 1042, 1056, 932, 1085,
    1092, 943, 1060, 1045, 958, 1072, 1068, 1043, 1050, 1066, 1037, 1082,
    1075, 1054, 929, 1089, 919, 923, 1069, 1080, 1046, 1078, 947, 1057, 1067,
    925)

# Calculate sample statistics
xbar1 <- mean(old_process)
n1 <- length(old_process)
s1 <- sd(old_process)
xbar2 <- mean(new_process)
n2 <- length(new_process)
s2 <- sd(new_process)

# Perform hypothesis test
Two_Samp_T_Test(xbar1, s1, n1, xbar2, s2, n2, alt = "left", mu = 0)
## Number of Degrees of Freedom: 61.4347 
## Test Statistic: t = -1.20701 
## P-Value: 0.11603




Two_Prop_Int

The Two_Prop_Int function calculates the confidence interval for the difference between two population proportions (p_1 - p_2).

 

Usage

The format is Two_Prop_Int(x1, n1, x2, n2, alpha) where x1 is the number of individuals of interest in Sample 1, n1 is the size of Sample 1, x2 is the number of individuals of interest in Sample 2, n2 is the size of Sample 2, and and alpha is a value between 0 and 1 where 1 - alpha is the confidence level as a decimal.

 

Return

The function will print the confidence level, margin of error, critical value, point estimate, and the confidence interval (lower and upper bounds).

Variables sprop1, sprop2, point_est, critval, Lbound, and Ubound are available as return values after execution of this function.

If invalid input values are given, an error message is returned.

 

Examples

Construct a 95% confidence interval for \(p_1 - p_2\) where \(x_1 = 74\), \(n_1 = 285\), \(x_2 = 168\), and \(n_2 = 400\). 

Two_Prop_Int(x1 = 74, n1 = 285, x2 = 168, n2 = 400, alpha = 0.05)
## Confidence Level: 95 % 
## Margin of Error: 0.07022 
## Critical Value: 1.95996 
## Point Estimate: -0.16035 
## Lower Bound: -0.23057 
## Upper Bound: -0.09013

Construct a 90% confidence interval for \(p_1 - p_2\) for the difference in the number of “Yes” in Sample 1 and Sample 2. 

# Define Sample 1
samp1 <- c("No", "Yes", "Yes", "No", "Yes", "Yes", "Yes", "No", "Yes", "Yes",
    "Yes", "Yes", "Yes", "Yes", "Yes", "Yes", "No", "Yes", "Yes", "Yes",
    "Yes", "Yes", "Yes", "Yes", "Yes", "Yes", "No", "Yes", "Yes", "Yes",
    "Yes", "Yes", "No", "Yes", "Yes", "Yes")

# Define Sample 2
samp2 <- c("No", "No", "Yes", "No", "No", "Yes", "No", "Yes", "Yes", "Yes",
    "No", "No", "No", "No", "Yes", "Yes", "Yes", "No", "Yes", "No", "No",
    "No", "Yes", "Yes", "Yes", "No", "No", "No", "Yes", "No", "No", "No",
    "No", "No", "No", "Yes", "No", "No", "Yes", "Yes")

# Summarize Sample 1 data into a table
table(samp1)
## samp1
##  No Yes 
##   6  30

# Summarize Sample 2 data into a table
table(samp2)
## samp2
##  No Yes 
##  24  16

# Extract Yes's, unname, and assign to x1 in Sample 1
x1 <- unname(table(samp1)["Yes"])

# Extract Yes's, unname, and assign to x2 in Sample 2
x2 <- unname(table(samp2)["Yes"])

# Determine sample size for Sample 1
n1 <- length(samp1)

# Determine sample size for Sample 2
n1 <- length(samp2)

# Explicitly state x1 (individuals of interest in Sample 1), n1 (Sample
# 1 size), x2 (individuals of interest in Sample 2), n2 (Sample 2
# size),
x1
## [1] 30
n1
## [1] 40
x2
## [1] 16
n2
## [1] 36

# Construct confidence interval
Two_Prop_Int(x1, n1, x2, n2, alpha = 0.1)
## Confidence Level: 90 % 
## Margin of Error: 0.17674 
## Critical Value: 1.64485 
## Point Estimate: 0.30556 
## Lower Bound: 0.12881 
## Upper Bound: 0.4823




Sign_Test

The Sign_Test function performs a one-sample Sign Test for a population median when the population distribution is not necessarily normal. Critical values are included based on specified alpha levels and alternate hypotheses.

 

Usage

The format is Sign_Test(sample, m0, alpha, alt) where sample is a numeric vector of the sample data, m0 is the hypothesized median, alpha is the significance level. Choices for one-tailed tests include 0.005, 0.01, 0.025, 0.05. Choices for two-tailed tests include 0.01, 0.02, 0.05, 0.10. alt is the form of the alternate hypothesis.

alt is a character string with choices “left”, “right”, or “two”. The choices “less”, “greater”, or “two.sided” are also accepted.

 

Return

The function will print detailed output including the null and alternate hypotheses, significance level, test statistic, critical value, and test result.

 

Examples

Perform a hypothesis test for \(H_0: m = 3\) versus \(H_1: m < 3\) at the \(\alpha = 0.05\) level of significance given a sample of data.

# Define sample data
data <- c(2.93, 2.95, 2.76, 2.89, 2.57, 3.06, 2.61, 2.66, 2.98, 2.79, 2.96,
    2.74)

# Perform the test
Sign_Test(data, m0 = 3, alpha = 0.05, alt = "less")
## Null Hypothesis: H0: m = 3 
## Alternate Hypothesis: H1: m < 3 
## Significance Level: 0.05 
## Test Statistic: 1 
## Critical Value: 2 
## Result: Reject the Null Hypothesis

Perform a hypothesis test for \(H_0: m = 170\) versus \(H_1: m < 170\) at the \(\alpha = 0.01\) level of significance given a sample of data.

# Define sample data
data <- c(149, 144, 218, 153, 134, 152, 148, 144, 178, 107, 199, 135, 171,
    110, 160, 119, 86, 127, 106, 153, 169, 153, 153, 173, 156, 145, 205,
    132, 169, 174, 130, 175)

# Perform the test
Sign_Test(data, m0 = 170, alpha = 0.01, alt = "less")
## Null Hypothesis: H0: m = 170 
## Alternate Hypothesis: H1: m < 170 
## Significance Level: 0.01 
## Test Statistic: -2.65165 
## Critical Value: -2.32635 
## Result: Reject the Null Hypothesis




Rank_Sum_Test

The Rank_Sum_Test function performs a nonparametric test for comparing the medians of two populations. The test requires that the populations have approximately the same shape, so the test is sometimes described as a test to determine whether two populations differ.

 

Usage

The format is Rank_Sum_Test(sample1, sample2, alpha, alt) where sample1 and sample2 are numeric vectors of the sample data, alpha is the significance level, alt is the form of the alternate hypothesis.

alt is a character string with choices “left”, “right”, or “two”. The choices “less”, “greater”, or “two.sided” are also accepted.

 

Return

The function will print detailed output including the null and alternate hypotheses, significance level, test statistic, p-value, and test result.

 

Examples

Decide whether there is a difference bewteen the median scores for two samples at the \(\alpha = 0.05\) level of significance.

# Define sample data
sample_1 <- c(78, 82, 83, 87, 75, 63, 78, 60, 94, 62, 98, 90, 97, 81)
sample_2 <- c(73, 72, 92, 100, 74, 90, 64, 84, 77, 89, 70, 64)

# Perform the test
Rank_Sum_Test(sample_1, sample_2, alpha = 0.05, alt = "two")
## Null Hypothesis: m1 = m2 
## Alternate Hypothesis: m1 is not equal to m2 
## Significance Level: 0.05 
## Test Statistic: -0.38576 
## P-Value: 0.69968 
## Result: Do Not Reject the Null Hypothesis




Signed_Rank_Test

The Signed_Rank_Test function performs the nonparametric Signed Rank Test for testing whether there is a difference between the medians of two populations, when the data are in the form of paired samples.

 

Usage

The format is Signed_Rank_Test(sample1, sample2, alpha) where sample1 and sample2 are numeric vectors of the sample data, and alpha is the significance level. Choices for alpha include 0.01, 0.02, 0.05, 0.10.

 

Return

The function will print detailed output including the null and alternate hypotheses, significance level, test statistic, critical value, and test result.

 

Examples

Decide whether there is a difference bewteen the median scores for two samples at the \(\alpha = 0.05\) level of significance.

# Define sample data
sample_1 <- c(283, 299, 274, 284, 248, 275, 293, 277)
sample_2 <- c(290, 281, 262, 287, 253, 287, 267, 271)

# Perform the test
Signed_Rank_Test(sample_1, sample_2, alpha = 0.05)
## Null Hypothesis: md = 0 
## Alternate Hypothesis: md is not equal to 0 
## Significance Level: 0.05 
## Test Statistic: 12.5 
## Critical Value: 4 
## Result: Do Not Reject the Null Hypothesis