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Chapter 11: Problem 7

The following data represent the measure of a variable before and after a treatment. $$\begin{array}{lccccc}\text { Individual } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\\\\hline \text { Before, } X_{i} & 93 & 102 & 90 & 112 & 107 \\\\\hline \text { After, } Y_{i} & 95 & 100 & 95 & 115 & 107 \\\\\hline\end{array}$$ Does the sample evidence suggest that the treatment is effective in increasing the value of the response variable? Use the \(\alpha=0.05\) level of significance. Note: Assume that the differenced data come from a population that is normally distributed with no outliers.

Short Answer

Expert verified
The treatment is effective based on the significance level of 0.05.

Step by step solution

01

- State the hypotheses

The null hypothesis (
02

Data Preparation

List the 'Before' and 'After' treatment measures.
03

Hypothesis Testing

Choose appropriate test for the provided data.
04

Conclusion

Interpret the results.

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Key Concepts

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

Paired Sample t-test
The paired sample t-test is a statistical method used to determine if there is a significant difference between the means of two related groups. In our case, the two related groups are the 'Before' and 'After' treatment measures for the same individuals. This test is particularly useful when you want to compare two conditions or time points for the same subjects. Here鈥檚 a step-by-step outline of how it works:
  • Calculate the difference between 'Before' and 'After' measures for each individual.
  • Find the mean and standard deviation of these differences.
  • Use these values to compute the t-statistic, which tells you how much the mean difference deviates from zero.
  • Compare the t-statistic to a critical value from the t-distribution table, based on your chosen level of significance and degrees of freedom (number of pairs minus one).
This test assumes that the differences between pairs are normally distributed. More about this important concept in our normal distribution section.
Level of Significance
Level of significance, often denoted as \(\alpha\), is a threshold set by the researcher which defines the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. In simpler terms, it determines how confident you want to be in your results before concluding that your findings are statistically significant.
  • Common levels of significance used are 0.05, 0.01, and 0.10. In this exercise, we use \(\alpha=0.05\), meaning we are willing to accept up to a 5% probability of making a Type I error.
  • If the p-value (calculated from your test statistic) is lower than your \(\alpha\), you reject the null hypothesis.
  • Conversely, if the p-value is higher, you do not reject the null hypothesis.
This helps you decide whether the observed effect is statistically significant or if it could have happened by random chance.
Normal Distribution
The normal distribution is a key concept in statistics, often referred to as the bell curve due to its shape. It is symmetric about its mean, with data values more concentrated around the mean and sparser the further away you go. For the paired sample t-test to be valid, the differences between pairs should be normally distributed. Here鈥檚 why normal distribution is important and some of its properties:
  • Symmetry: The left and right halves of a normal distribution are mirror images of each other.
  • Central Peak: The highest point on the curve is the mean, median, and mode of the data.
  • Empirical Rule: Approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
In practice, if you plot the differences between 'Before' and 'After' measures and they form this bell shape, your data likely follows a normal distribution. This ensures that the assumptions of the paired sample t-test are met, making your results valid and reliable.

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Most popular questions from this chapter

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