An experiment was designed to estimate the mean difference...

An experiment was designed to estimate the mean difference in weight gain for pigs fed ration $\mathrm{A}$ as compared with those fed ration B. Eight pairs of pigs were used. The pigs within each pair were littermates. The rations were assigned at random to the two animals within each pair. The gains (in pounds) after 45 days are shown in the table at top of next column.Assuming weight gain is normal, find the $95 \%$ confidence interval estimate for the mean of the differences $\mu_{d}$ where $d=$ ration $\mathrm{A}-$ ration $\mathbf{B}$.

Key Concepts

Normal Distribution Assumption

The normal distribution assumption means that the differences between paired observations are assumed to be normally distributed. This assumption is necessary for the validity of the paired t-test and the corresponding confidence interval calculations, particularly when the sample size is small.

Confidence Interval

A confidence interval provides a range of plausible values for an unknown parameter, such as a mean difference, and is constructed so that there is a specified probability (e.g., 95%) that the interval contains the true parameter value. In the context of a paired t-test, the confidence interval is built from the sample mean difference and its estimated standard error.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that are free to vary in a calculation. In a paired t-test with n pairs, the degrees of freedom are typically (n - 1). This concept is critical because it affects the shape of the t-distribution used to calculate the confidence interval and p-values.

Paired Design

A paired design is used when subjects or experimental units are naturally paired or matched in the study, such as in twin studies or in controlled experiments where each subject experiences both conditions. This design helps to control for variability between pairs by ensuring that each comparison is made within closely related or identical units, thus isolating the treatment effect.

Paired t-Test

The paired t-test is a statistical method used to compare the means of two related groups. It is specifically applied when the data consist of paired observations, and the differences within each pair are analyzed. This test accounts for the pairing by reducing the impact of between-subject variability, thus providing a more sensitive analysis of the treatment effect.

Transcript

00:01 All right, let's look at question 19 together.

00:04 So by the definition of d given in the question, d value is equal to ration a minus ration b.

00:13 So now we're going to first find out each value for d respect to each subject.

00:23 So for the first subject, the d is 65 minus 58, so it's 7, and the second is negative 2, 9, 2, 2, 10, negative 6, and 8.

00:47 All right.

00:50 Now we're going to calculate for the d squared as well because we will use the value for d squared in the following steps.

01:04 So the d square for 7 is 49, 481, 4, 4, 4, 1, 136, and 64.

01:18 All right.

01:22 And notice that our sample size here is 8.

01:26 So n is 8.

01:28 And because our confidence level is 95%, so our alpha is equal to 0 .05.

01:37 All right.

01:39 Now we can calculate summation of d, which is 30.

01:51 Is equal to 342.

01:58 And in where like if we want to calculate for the standard deviation of d, we also need to know the square of summation d, which is square of 30, and it is equal to 900.

02:19 All right, by having this much information, we'll be able to calculate center deviation for d.

02:27 I'll just write down the formula.

02:34 So it is summation of d squared minus square of summation d divided by n over n minus 1.

02:48 Alright, so we just plug in these numbers, 342 minus 900 over 8 divided by 8 minus 1...

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