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To get critical values of the z-score in R for one-tailed and two-tailed tests, you can use the qnorm() function.
For a one-tailed test with a significance level of α, the critical value is the z-score that corresponds to a cumulative probability of 1-α in the standard normal distribution.
For example, to get the critical value of the z-score for a one-tailed test with a significance level of 0.05 (i.e., α=0.05) in R, you can use the following code:
# One-tailed test with significance level of 0.05 (to the right)
critical_value <- qnorm(1-0.05, lower.tail=TRUE)
critical_value[1] 1.644854# One-tailed test with significance level of 0.05 (to the left)
critical_value <- qnorm(0.05, lower.tail=TRUE)
critical_value[1] -1.644854## Alternatively
critical_value <- qnorm(1-0.05, lower.tail=FALSE)
critical_value[1] -1.644854# Two-tailed test with significance level of 0.05
critical_value1 <- qnorm(0.025, lower.tail=TRUE)
critical_value2 <- qnorm(0.025, lower.tail=FALSE)
critical_value1[1] -1.959964critical_value2[1] 1.959964p1 = 0.232
n1 = 3000
p2 = 0.267
n2 = 3000
## p_hat = (x1 + x2) / (n1 + n2)
p_hat = (0.232*3000 + 0.267*3000) / (3000 + 3000)
z = (p1 - p2) / sqrt(p_hat * (1 - p_hat) * (1/n1 + 1/n2))
z[1] -3.132586x <- c(34,45,30,50,36,45)
y <- c(30,33,40,34,40,54)# Calculate the z-score standardized for the difference in means
z_score <- (mean(x) - mean(y)) / sqrt(var(x)/length(x) + var(y)/length(y))
z_score[1] 0.317524x_bar <- 45
mu <- 48
n <- 50
sig <- 12
zscore <- ((x_bar - mu)/(sig/sqrt(n)))
zscore[1] -1.767767# One-tailed test with z-score of 1.5
p_value <- pnorm(1.5, lower.tail=FALSE)
p_value[1] 0.0668072# Two-tailed test with z-score of -2.0
p_value <- 2*pnorm(-2.0, lower.tail=TRUE)
p_value[1] 0.04550026To get the critical values for a t-test in R, we can use the qt() function. The qt() function takes three arguments: p, df, and lower.tail. The p argument specifies the probability of the tail, df specifies the degrees of freedom, and lower.tail specifies whether to calculate the critical value for the lower tail or upper tail of the t-distribution.
If we want to find the t critical value for a left-tailed test with a significance level of .05 and degrees of freedom = 22, we can use the following example code
#find t critical value
qt(p=.05, df=22, lower.tail=TRUE)[1] -1.717144The t critical value for a left-tailed test is -1.7171. Similarly, if we want to find the t critical value for a right-tailed test with a significance level of .05 and degrees of freedom = 22, we can use the following example code
#find t critical value
qt(p=.05, df=22, lower.tail=FALSE)[1] 1.717144If we want to find the t critical values for a two-tailed test with a significance level of .05 and degrees of freedom = 22, we can use the following example code
#find two-tailed t critical values
qt(p=.05/2, df=22, lower.tail=FALSE)[1] 2.073873For example, if we want to find the p-value associated with a t-score of -0.77 and df = 15 in a left-tailed hypothesis test, we can use the following code:
#find p-value
pt(q=-.77, df=15, lower.tail=TRUE)[1] 0.2266283# find two-tailed p-value
p_value <- 2 * pt(q = abs(2.5), df = 10, lower.tail = FALSE)
p_value[1] 0.03144684A researcher wants to test whether there is a significant difference in the mean scores of two groups of students (Group A and Group B) on a math test. The mean score for Group A is 85 with a standard deviation of 6, while the mean score for Group B is 90 with a standard deviation of 7. The sample size for both groups is 30. The researcher decides to use a significance level of α = 0.05.
What is the calculated t-value and p-value for this experiment?
x1 = 85
x2 = 90
s1 = 6
s2 = 7
n1 = 30
n2 = 30t = (x1 - x2) / (s1 /sqrt(n1)) + (s1/sqrt(n2))
t[1] -3.46891qt(p=.05/2, df=58, lower.tail=FALSE)[1] 2.001717p_value <- 2 * pt(q = abs(-3.46891), df = 58, lower.tail = FALSE)
p_value[1] 0.0009920902This brings us to the same decision.
# Create a vector of IDs
id <- 1:20
# Create a vector of weights before
weight_before <- c(60, 65, 70, 72, 68, 74, 80, 76, 62, 68, 75, 79, 72, 70, 67, 63, 65, 71, 73, 69)
# Create a vector of weights after
weight_after <- c(61, 64, 72, 70, 70, 75, 81, 78, 63, 67, 74, 80, 73, 71, 68, 64, 66, 72, 75, 71)
# Create a data frame with the three vectors
weight_data <- data.frame(id, weight_before, weight_after)
head(weight_data,5) id weight_before weight_after
1 1 60 61
2 2 65 64
3 3 70 72
4 4 72 70
5 5 68 70attach(weight_data)To calculate the dependent t-test manually, you can follow these steps:
weight_data$dif <-weight_before-weight_after
head(weight_data,5) id weight_before weight_after dif
1 1 60 61 -1
2 2 65 64 1
3 3 70 72 -2
4 4 72 70 2
5 5 68 70 -2mean_dif <- mean(weight_data$dif)
sd_dif <- sd(weight_data$dif)se_dif <- sd_dif / sqrt(length(weight_data$dif))
se_dif[1] 0.2575185t_stat <- mean_dif / se_dif
t_stat[1] -3.106573dof <- length(weight_data$dif) - 1
dof[1] 19qt(p=.05/2, df=19, lower.tail=FALSE)[1] 2.093024# find two-tailed p-value
p_value <- 2 * pt(q = abs(-3.106573), df = 19, lower.tail = FALSE)
p_value[1] 0.005809366lower <- mean_dif - qt(0.025, df = dof) * se_dif
upper <- mean_dif + qt(0.025, df = dof) * se_dif
lower[1] -0.2610075upper[1] -1.338992