Was used to complete a t -test of the difference between two...

Was used to complete a $t$ -test of the difference between two means using the following two independent samples. $$\begin{array}{lllllllll}\hline \text { Sample 1 } & 33.7 & 21.6 & 32.1 & 38.2 & 33.2 & 35.9 & 34.1 & 39.8 \\& 23.5 & 21.2 & 23.3 & 18.9 & 30.3 & & & \\\hline \text { Sample 2 } & 28.0 & 59.9 & 22.3 & 43.3 & 43.6 & 24.1 & 6.9 & 14.1 \\& 30.2 & 3.1 & 13.9 & 19.7 & 16.6 & 13.8 & 62.1 & 28.1 \\\hline\end{array}$$ a. Assuming normality, verify the results (two sample means and standard deviations, and the calculated $t \star)$ by calculating the values yourself. b. Use Table 7 in Appendix $B$ to verify the $p$ -value based on the calculated df. c. Find the $p$ -value using the smaller number of degrees of freedom. Compare the two $p$ -values.

Was used to complete a $t$ -test of the difference between two means using the following two independent samples. $$\begin{array}{lllllllll}\hline \text { Sample 1 } & 33.7 & 21.6 & 32.1 & 38.2 & 33.2 & 35.9 & 34.1 & 39.8 \\& 23.5 & 21.2 & 23.3 & 18.9 & 30.3 & & & \\\hline \text { Sample 2 } & 28.0 & 59.9 & 22.3 & 43.3 & 43.6 & 24.1 & 6.9 & 14.1 \\& 30.2 & 3.1 & 13.9 & 19.7 & 16.6 & 13.8 & 62.1 & 28.1 \\\hline\end{array}$$
a. Assuming normality, verify the results (two sample means and standard deviations, and the calculated $t \star)$ by calculating the values yourself.
b. Use Table 7 in Appendix $B$ to verify the $p$ -value based on the calculated df.
c. Find the $p$ -value using the smaller number of degrees of freedom. Compare the two $p$ -values.

Key Concepts

p-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one computed from the sample data, under the assumption that the null hypothesis is true. It provides a measure of the strength of evidence against the null hypothesis; a smaller p-value indicates stronger evidence to reject the null hypothesis in favor of the alternative.

Degrees of Freedom

Degrees of freedom in the context of a t-test refer to the number of independent values that are free to vary when estimating statistical parameters. They are used in determining the appropriate t-distribution for the test statistic and are essential for accurately assessing the significance of the test by matching the calculated t-value to a critical value or p-value from the relevant distribution.

Normality Assumption

The normality assumption is the premise that the data in each sample are drawn from a normally distributed population. This assumption is important for many statistical tests, including the two-sample t-test, because the validity and accuracy of the t-distribution approximation depend on the underlying data being approximately normally distributed, particularly for small sample sizes.

Sample Mean

The sample mean is the average of a set of observations and is a measure of central tendency. It is calculated by summing all the values in the sample and dividing by the number of observations. In hypothesis testing, sample means are used to estimate the population means and to assess the magnitude of differences between groups.

Two-sample t-test

A two-sample t-test is a statistical procedure used to compare the means of two independent groups to determine if there is a statistically significant difference between them. It involves calculating a t-statistic based on the difference between the sample means relative to the variability within the groups, and it assumes that the data are normally distributed and that the variances of the populations are equal or approximately equal.

Sample Standard Deviation

The sample standard deviation quantifies the amount of variation or dispersion in a set of sample data. It is calculated by taking the square root of the variance, which is the average of the squared differences from the sample mean. This statistic is crucial in a t-test as it normalizes the difference between sample means by accounting for the spread of the data.

Transcript

00:01 In this problem, we are asked to verify the sample means, sample standard deviations, and test statistic presented in the minutab output by calculating them ourselves.

00:10 For the sample means and sample standard deviations, i'm using excel.

00:15 So i've copied the sample information from the question and put it into excel.

00:20 So we're ready to calculate the sample means and sample standard deviations.

00:25 So for the means, we can use the average function.

00:35 And for the sample standard deviations, we can use the st, de, ve.

00:38 V .s function.

00:50 And we have the two sample means and the two sample standard deviations.

00:54 And you can verify that these are the same as the values in the mini tab output.

00:59 So what those verified, the third thing to verify is the test statistic.

01:08 And that is given by this formula.

01:09 It's the difference in sample means divided by the standard error.

01:51 And this comes out to 0 .591, which is in accordance with the test statistic from the mini tab output, which is 0 .59.

02:00 We've just gone to three decimal places instead of.

02:07 And for part b, we're asked to use table 7 at the back of the book to verify the p value based on the calculated degrees of freedom.

02:14 And the calculated degrees of freedom is that calculated by mini tab in the output.

02:19 So that's degrees of freedom is equal to 20.

02:22 It's from the question.

02:26 So if we use a degrees of freedom equal to 20, we're looking in somewhere in between these two columns.

02:36 And our test statistic is 0 .59, so it's between these two rows.

02:41 So we're looking at a p -value range between these four cells...

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