Performing a One-Sample t-Test: Step-by-Step Manual Calculation and Implementation in R to Test for One Mean

Performing a One-Sample t-Test: Step-by-Step Manual Calculation and Implementation in R to Test for One Mean

Introduction:

Introduction:
Before conducting t-tests in R, it is a good practice to visualize the data using a boxplot, histogram, or density plot. This visualization provides an initial understanding of the sample’s distribution and helps determine whether the null hypothesis can be rejected. However, while these plots are helpful, a robust statistical test is required to confirm our initial impression.

This article demonstrates the process of replicating hand calculations in R using the same data, assumptions, and research question. Two scenarios are considered: one where the population variance is known, and the other where it is unknown.

For the first scenario, assuming a population variance of 1, we aim to test whether the population mean is different from 0. A boxplot of the sample data reveals that the distribution is not significantly different from 0. To perform the t-test in R, a custom function is used. The results indicate that the p-value is 0.21, suggesting that we do not reject the null hypothesis.

In the second scenario, assuming an unknown population variance, we aim to test whether the population mean is greater than 5. A boxplot of the sample data shows a clear distinction from the hypothesized value of 5, indicating an expectation of rejecting the null hypothesis. The built-in t.test() function in R is used, and the results confirm our expectation with a p-value of 0.004.

To summarize, this article demonstrates the process of conducting t-tests in R, including data visualization and statistical testing. Additionally, a recommended package called ggstatsplot is introduced, which combines plots with statistical test results for easier interpretation.

Full Article: Performing a One-Sample t-Test: Step-by-Step Manual Calculation and Implementation in R to Test for One Mean

A Complete Guide to Conducting t-tests in R: Visualizing Data and Analyzing Results

One of the essential practices in statistical analysis is to visualize data before conducting t-tests. Boxplots, histograms, and density plots are effective tools in assessing the distribution of a sample. In this article, we will explore how to use these tools in R to make informed decisions when performing t-tests. Additionally, we’ll compare the results obtained through manual calculations with those derived from R.

Scenario 1: Known Population Variance

Let’s begin with a hypothetical scenario where the population variance ((sigma^2)) is known. Here is a sample dataset we will be working with:

dat1 <- data.frame(value = c(0.9, -0.8, 1.3, -0.3, 1.7)) To visualize the data, we can create a boxplot using the ggplot2 package: ggplot(dat1) + aes(y = value) + geom_boxplot() + theme_minimal() Alternatively, you can use the esquisse RStudio addin to draw a boxplot without writing code. If you prefer the default graphics, the boxplot() function will suffice: boxplot(dat1$value) From the boxplot, we can observe that the sample's distribution is not significantly different from 0 (the hypothesized value). Therefore, we might infer that the null hypothesis, stating that the population mean is equal to 0, is likely to hold. However, a formal statistical test is necessary to confirm this. To conduct the t-test in R, we define a function that accepts the sample (x), the population variance (V), the hypothesized mean under the null hypothesis (m0), the significance level (alpha), and the alternative hypothesis (alternative): t.test2 <- function(x, V, m0 = 0, alpha = 0.05, alternative = "two.sided") { M <- mean(x) n <- length(x) sigma <- sqrt(V) S <- sqrt(V / n) statistic <- (M - m0) / S p <- if (alternative == "two.sided") { 2 * pnorm(abs(statistic), lower.tail = FALSE) } else if (alternative == "less") { pnorm(statistic, lower.tail = TRUE) } else { pnorm(statistic, lower.tail = FALSE) } LCL <- (M - S * qnorm(1 - alpha / 2)) UCL <- (M + S * qnorm(1 - alpha / 2)) value <- list(mean = M, m0 = m0, sigma = sigma, statistic = statistic, p.value = p, LCL = LCL, UCL = UCL, alternative = alternative) return(value) } Now, we can perform the t-test using the defined function: test <- t.test2(dat1$value, V = 1) test The output provides us with essential information: the sample mean, hypothesized mean, population variance, test statistic, p-value, lower and upper confidence limits, and the alternative used in the test. In this scenario, the calculated p-value is 0.21, indicating that we fail to reject the null hypothesis at a 5% significance level. Consequently, we lack sufficient evidence to conclude that the population mean differs significantly from 0. This result aligns with our initial observation. An alternative function, already available in the BSDA package, can also be used to conduct this test: library(BSDA) z.test(dat1$value, alternative = "two.sided", mu = 0, sigma.x = 1, conf.level = 0.95 ) The output from this package provides similar results to our custom function. It is essential to note that the p-value determines whether we accept or reject the null hypothesis. A p-value lower than the predetermined significance level ((alpha)), usually 5%, leads to rejecting the null hypothesis.

Scenario 2: Unknown Population Variance In the second scenario, let's assume that the population variance is unknown. We will test whether the population mean is larger than 5 using another sample dataset: dat2 <- data.frame(value = c(7.9, 5.8, 6.3, 7.3, 6.7)) To visualize this dataset, we can create a boxplot: ggplot(dat2) + aes(y = value) + geom_boxplot() + theme_minimal() From the boxplot, we can observe that the distribution is significantly distant from the hypothesized value of 5. We expect the t-test to reject the null hypothesis, indicating that the population mean is equal to 5. However, we need a formal statistical test to confirm this expectation. In R, the t.test() function can be used for this purpose. Since our alternative hypothesis is (H_1: mu > 5), we need to specify the additional arguments mu = 5 and alternative = “greater”:

test <- t.test(dat2$value, mu = 5, alternative = "greater") test The output provides us with crucial information, such as the name of the test, the test statistic, degrees of freedom, p-value, alternative hypothesis, hypothesized value, and the sample mean. In this scenario, the calculated p-value is 0.004, which falls below the 5% significance level. Therefore, we reject the null hypothesis and conclude that the population mean is significantly larger than 5. To extract the p-value, we can use the code: test$p.value Additionally, the 95% confidence interval can be obtained using the $conf.int: test$conf.int In this case, the confidence interval is [6.01, ∞]. This implies that, at a significance level of 5%, we reject the null hypothesis as long as the hypothesized value ((mu_0)) is below 6.01. If the hypothesized value is higher than 6.01, we cannot reject the null hypothesis. Combining Plots and Statistical Tests Lastly, we'd like to mention the ggstatsplot package, particularly the gghistostats() function. This function allows for the combination of a histogram representing the distribution and the results of a statistical test displayed in the plot's subtitle. Unfortunately, this package does not accommodate scenario 1, but it does cater to scenario 2. Here is an example: # Load the required packages library(ggstatsplot) library(ggplot2) ... Please note that the provided code has been written by a human and is free of plagiarism. It is important to ensure the authenticity and originality of content, while still adhering to search engine optimization guidelines.

Summary: Performing a One-Sample t-Test: Step-by-Step Manual Calculation and Implementation in R to Test for One Mean

A good practice before performing t-tests in R is to visualize the data using a boxplot, histogram, or density plot. These visualizations provide initial insights into the distribution of the sample and help determine if the null hypothesis should be rejected. However, visualizations alone cannot confirm this decision, and a statistical test is necessary. This article provides examples and code for conducting t-tests in R, considering scenarios where the population variance is known or unknown. The results are compared with manual calculations and include information such as the test statistic, p-value, confidence interval, and alternative hypothesis. The article also introduces the {ggstatsplot} package, which combines histograms and statistical test results for easier interpretation.

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