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The following data are taken from four different populations that are known to be normally distributed, with equal population variances based on independent simple random samples. $$ \begin{array}{cccc} \text { Sample 1 } & \text { Sample 2 } & \text { Sample 3 } & \text { Sample 4 } \\ \hline 110 & 138 & 98 & 130 \\ \hline 85 & 140 & 100 & 116 \\ \hline 83 & 130 & 94 & 157 \\ \hline 95 & 115 & 110 & 137 \\ \hline 103 & 101 & 104 & 144 \\ \hline 105 & 130 & 118 & 124 \\ \hline 107 & 123 & 102 & 139 \\ \hline \end{array} $$ (a) Test the hypothesis that each sample comes from a population with the same mean at the \(\alpha=0.05\) level of significance. That is, test \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}\). (b) If you rejected the null hypothesis in part (a), use Tukey's test to determine which pairwise means differ using a familywise error rate of \(\alpha=0.05\). (c) Draw boxplots of each set of sample data to support your results from parts (a) and (b).

Step by step solution

01

State the Hypotheses

We are testing the null hypothesis that all populations have the same mean. Set up the hypotheses as follows: Null hypothesis: \( H_{0}: \mu_{1} = \mu_{2} = \mu_{3} = \mu_{4} \)Alternative hypothesis: \( H_{a}: \mu_{i} \text{ are not all equal} \)

02

Calculate Sample Means and Variances

Calculate the mean and variance for each sample set. This is done by summing all the values in each sample and dividing by the number of observations (mean) and then computing the variance.

03

Conduct ANOVA Test

Use analysis of variance (ANOVA) to test whether there are any statistically significant differences between the means of the independent groups. Compute the F-statistic and compare it to the critical value from the F-distribution table with the appropriate degrees of freedom at \( \alpha = 0.05 \) significance level.

04

Make Decision Regarding Hypothesis

If the computed F-statistic is greater than the critical value, reject the null hypothesis. Otherwise, do not reject the null hypothesis.

05

Perform Tukey's HSD Test (if required)

If the null hypothesis was rejected in Step 4, perform Tukey's Honestly Significant Difference (HSD) test to determine which means are statistically different. This involves calculating the HSD value and comparing the differences of each pair of sample means against this value.

06

Draw Boxplots

Draw boxplots for each sample data to visually inspect the center and spread of the data. Check whether the boxplots support the findings from the ANOVA and Tukey's HSD test.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

ANOVA

Analysis of Variance (ANOVA) is a powerful statistical technique used to compare the means of three or more groups. It helps determine if at least one of the group means is different from the others. Imagine you are looking at four different groups of data, like in our given example.

To perform ANOVA, we first set up our hypotheses:

• **Null Hypothesis (H鈧�)**: All group means are equal.
• **Alternative Hypothesis (H鈧�)**: At least one group mean is different.

By computing the F-statistic, which assesses the ratio of between-group variance to within-group variance, we can infer if the observed differences are statistically significant. If the F-statistic is higher than the critical value from the F-distribution table at a chosen significance level (e.g., \( \alpha=0.05 \)), we reject the null hypothesis. This paves the way for further analysis.

Tukey's HSD Test

Once you've determined through ANOVA that there is a significant difference among group means, you might want to know *which* specific groups differ. This is where Tukey's Honestly Significant Difference (HSD) test comes in.

Tukey's HSD test compares all possible pairs of means to identify those that are significantly different. It controls the familywise error rate, ensuring that the probability of making one or more type I errors is minimized.

The test calculates a critical value, called the HSD value, and then compares the actual observed mean differences to this critical value. If the difference between any two groups is larger than the HSD value, then we conclude those groups have significantly different means. This helps drill down to see exactly where the differences lie among your groups.

Boxplots

Boxplots, also known as whisker plots, are an excellent graphical tool for summarizing the distribution of a dataset. They can show you the central tendency, variability, and potential outliers in a simple format.

In our exercise, drawing boxplots for each of the four samples provides a visual support for the results from ANOVA and Tukey's HSD test.

Each boxplot displays:

• The median (middle line inside the box).
• The interquartile range (IQR), shown by the box itself, which represents the middle 50% of the data.
• The whiskers, which extend to the smallest and largest values within 1.5*IQR from the quartiles.

By comparing the boxplots side-by-side, you can visually inspect if there are noticeable differences between the group means and variances. This often gives a clear picture complementing your analytical results.

Independent Samples

In our exercise, we deal with independent samples. This means the samples are collected in such a way that the selection of one sample doesn鈥檛 influence the others. Independent sampling is crucial for the validity of ANOVA and other statistical tests.

Imagine picking random apples from four different baskets. The apple you pick from one basket does not affect your pick from another basket. This independence ensures that comparisons among group means are valid and not biased by any inherent connections between samples.

Ensuring independence helps us rely on the results of statistical tests like ANOVA and Tukey's HSD test without worrying about inter-sample dependencies skewing the outcomes.

Normal Distribution

One of the assumptions for ANOVA and Tukey's HSD test is that the populations from which samples are drawn are normally distributed.

A normal distribution is often depicted as the classic bell-shaped curve, symmetric around the mean. This shape of the distribution is important because many statistical tests rely on it. So how do we know if our data is normally distributed?

Methods like:

• Visual inspection using histograms or Q-Q plots.
• Statistical tests like Shapiro-Wilk or Anderson-Darling.

Evaluating the normality of distributions helps ensure that the assumptions for ANOVA hold true. If the data is not normally distributed, transformations or non-parametric methods might be considered to achieve accurate results.