The independent samples z-test is used to compare the means of two independent groups to determine if they are significantly different from each other. It is based on the assumption that the population standard deviations of both groups are known.
Let’s consider two independent groups, Group A and Group B. We want to test if the mean of Group A \((\mu_A)\) is different from the mean of Group B \((\mu_B)\). The null and alternative hypotheses are defined as follows:
Let’s consider the weights of two independent groups of participants. Group A consists of 50 individuals with a mean weight of 70 kg and a known population standard deviation of 5 kg. Group B consists of 60 individuals with a mean weight of 65 kg and a known population standard deviation of 4 kg. We want to test if there is a significant difference in the mean weights of the two groups.
We have two independent groups, Group A and Group B, with the following characteristics:
We want to test if there is a significant difference in the mean weights of the two groups.
Let’s set \(\alpha = 0.05\) (95% confidence level).
The test statistic for the independent samples z-test is given by:
\(z = \frac{{\bar{X}_A - \bar{X}_B}}{{\sqrt{{\frac{{\sigma_A^2}}{{n_A}} + \frac{{\sigma_B^2}}{{n_B}}}}}}\)
Where: - \(\bar{X}_A\) = mean weight of Group A - \(\bar{X}_B\) = mean weight of Group B - \(\sigma_A\) = population standard deviation of Group A - \(\sigma_B\) = population standard deviation of Group B - \(n_A\) = sample size of Group A - \(n_B\) = sample size of Group B
Let’s calculate the test statistic:
X_A <- 70
X_B <- 65
sigma_A <- 5
sigma_B <- 4
n_A <- 50
n_B <- 60
z <- (X_A - X_B) / sqrt((sigma_A^2 / n_A) + (sigma_B^2 / n_B))
z[1] 5.710402The test statistic (z) is calculated as 5.7104024.
For a two-tailed test with a significance level of \(\alpha = 0.05\), the critical value is obtained from the standard normal distribution. We need to find the value of z that corresponds to an area of \(\frac{{\alpha}}{2}\) in each tail.
critical_value <- qnorm(1 - 0.05/2)
critical_value[1] 1.959964The critical value is approximately 1.959964.
We compare the absolute value of the test statistic (|z|) with the critical value to make our decision.
if (abs(z) > critical_value) {
conclusion <- "Reject the null hypothesis."
} else {
conclusion <- "Fail to reject the null hypothesis."
}
conclusion[1] "Reject the null hypothesis."Since the test statistic (5.7104024) is greater than the critical value (1.959964), we reject the null hypothesis.
X_A <- 70
X_B <- 65
sigma_A <- 5
sigma_B <- 4
n_A <- 50
n_B <- 60
z <- (X_A - X_B) / sqrt((sigma_A^2 / n_A) + (sigma_B^2 / n_B))
critical_value <- qnorm(1 - 0.05/2)
z[1] 5.710402The calculated test statistic (z = 5.7104024) is greater than the critical value (1.96). Hence, we reject the null hypothesis.
Let’s use the same significance level as before: \(\alpha = 0.05\).
We have already calculated the test statistic (z) in the previous method.
The p-value represents the probability of obtaining a test statistic as extreme as the observed value (or more extreme), assuming the null hypothesis is true.
The calculated p-value is very small (p < 0.001).
We compare the p-value with the significance level (\(\alpha\)) to make our decision.
if (p_value <= 0.05) {
conclusion <- "Reject the null hypothesis."
} else {
conclusion <- "Fail to reject the null hypothesis."
}
conclusion[1] "Reject the null hypothesis."Since the p-value (p < 0.001) is less than the significance level (0.05), we reject the null hypothesis.
The calculated p-value (p = 1.1270936^{-8}) is less than the significance level (\(\alpha = 0.05\)). Hence, we reject the null hypothesis.
Let’s use a 95% confidence level, which corresponds to \(\alpha = 0.05\).
We have already calculated the test statistic (z) in the previous methods.
For a 95% confidence level, we need to find the critical value from the standard normal distribution.
critical_value <- qnorm(1 - 0.05/2)
critical_value[1] 1.959964The critical value is approximately 1.96.
The confidence interval for the difference in means is given by:
\((\bar{X}_A - \bar{X}_B) \pm z \times \sqrt{{\frac{{\sigma_A^2}}{{n_A}} + \frac{{\sigma_B^2}}{{n_B}}}}\)
lower <- (X_A - X_B) - critical_value * sqrt((sigma_A^2 / n_A) + (sigma_B^2 / n_B))
upper <- (X_A - X_B) + critical_value * sqrt((sigma_A^2 / n_A) + (sigma_B^2 / n_B))
lower[1] 3.283865upper[1] 6.716135The confidence interval is (3.2838653, 6.7161347).
If zero is not within the confidence interval, we reject the null hypothesis.
if (lower <= 0 & upper >= 0) {
conclusion <- "Fail to reject the null hypothesis."
} else {
conclusion <- "Reject the null hypothesis."
}
conclusion[1] "Reject the null hypothesis."Since zero (representing no difference in means) falls within the confidence interval (3.2838653, 6.7161347), we fail to reject the null hypothesis.
lower <- (X_A - X_B) - critical_value * sqrt((sigma_A^2 / n_A) + (sigma_B^2 / n_B))
upper <- (X_A - X_B) + critical_value * sqrt((sigma_A^2 / n_A) + (sigma_B^2 / n_B))
lower[1] 3.283865upper[1] 6.716135The confidence interval (3.2838653, 6.7161347) do not includes zero, which indicates that there is significant difference between the means of Group A and Group B. Hence, we reject the null hypothesis.
In this document, we discussed the independent samples z-test and explained how to perform hypothesis testing using the critical value method, p-value method, and confidence interval method. Each method provides a way to assess whether the means of two independent groups are significantly different. It is essential to understand and apply these methods correctly to make valid conclusions based on the available sample data.
In this analysis, we tested if there is a significant difference in the mean weights of two independent groups using three methods: the critical value method, p-value method, and confidence interval method. The results consistently led to the conclusion that there is a significant difference in the mean weights of the two groups. These methods provide different approaches to hypothesis testing, but in this case, they all yielded the same conclusion.
It is important to consider the assumptions and limitations of the independent samples z-test when applying it to real-world scenarios. Additionally, collecting larger sample sizes and using random sampling techniques can help improve the reliability of the results.