Question

In a study of the nutritional requirements of cattle, researchers measured the weight gains of cows during a 78 -day period. For two breeds of cows, Hereford (HH) and Brown Swiss/Hereford (SH), the results are summarized in the following table. \({ }^{6}\) $$ \begin{array}{|lll|} \hline & \text { HH } & \text { SH } \\ \hline n & 33 & 51 \\ \bar{y} & 18.3 & 13.9 \\ s & 17.8 & 19.1 \\ \hline \end{array} $$ (a) What is the value of the \(t\) test statistic for comparing the means? (b) In the context of this study, state the null and alternative hypotheses. (c) The \(P\) -value for the \(t\) test is \(0.29 .\) If \(\alpha=0.10\), what is your conclusion regarding the hypotheses in (b)?

Short answer

The value of the t-test statistic is found using the means, standard deviations, and sample sizes given. If the P-value (0.29) is greater than the significance level (0.10), the null hypothesis is not rejected.

Step-by-step solution

Understanding the data

Review and understand the given information. In this case, for each breed (HH and SH), we have the number of cows (\(n\)), the mean weight gain (\bar{y}\

Calculate the standard error of the difference in means

The standard error of the difference in means is found using the formula: \(SE = \sqrt{\frac{s_{HH}^2}{n_{HH}} + \frac{s_{SH}^2}{n_{SH}}}\). Substitute the given standard deviations and sample sizes for each cow breed and calculate the value of SE.

Calculate the t test statistic

Use the formula \(\(t = \frac{\bar{y}_{HH} - \bar{y}_{SH}}{SE}\)\) to calculate the value of the t-test statistic by substituting the mean weight gains and the standard error calculated in Step 2.

State the null and alternative hypotheses

Write down the null hypothesis (\(H_0\) which posits no difference in means between the two populations) and the alternative hypothesis (\(H_A\) which suggests there is a significant difference).

Make a decision regarding the hypotheses

Compare the P-value to the significance level \(\alpha=0.10\) to decide whether to reject the null hypothesis. If P-value > \alpha, do not reject the null; if P-value < \alpha, reject the null.

Key concepts

Null and Alternative Hypotheses

When we perform a statistical test, our starting point is a set of hypotheses. In a t-test, just like in the given exercise where researchers measure weight gains in cows, we begin by stating the null hypothesis (\(H_0\)), which is a statement of no effect or no difference. In this scenario, our null hypothesis would be that there is no significant difference in the mean weight gain between Hereford cows (HH) and Brown Swiss/Hereford cows (SH). Mathematically, we'd say \(H_0: \bar{y}_{HH} = \bar{y}_{SH}\).
The alternative hypothesis (\(H_A\)), on the other hand, is what we're trying to provide evidence for — it's the opposite of the null. In the exercise, the alternative hypothesis asserts that there is a significant difference in the weight gains of the two breeds, which we express as \(H_A: \bar{y}_{HH} eq \bar{y}_{SH}\).
Understanding these hypotheses is crucial because they guide the direction of the statistical testing and influence the interpretation of the results. The null hypothesis is like a courtroom's presumption of innocence; the data collected needs to provide enough evidence to 'convict', or in this case, to reject the null hypothesis.

P-value Interpretation

A P-value is a probability that measures the evidence against the null hypothesis. Lower P-values indicate stronger evidence in favor of the alternative hypothesis. In the context of our cow weight gain study, the P-value is 0.29. To interpret this, compare it with the significance level, commonly denoted as \(\alpha\). The significance level represents the risk we are willing to take of incorrectly rejecting a true null hypothesis (a type I error).
Here, if our significance level \(\alpha = 0.10\), it means we accept a 10% chance of making a type I error. Since 0.29 is greater than 0.10, we do not have sufficient evidence to reject the null hypothesis. Hence, we would conclude that there is not enough statistical evidence to say that the mean weight gain for the two cow breeds is significantly different; in the judicial metaphor, the null hypothesis cannot be convicted with the available evidence.

Standard Error Calculation

The standard error (SE) is a measure of how much we expect our sample mean to vary from the true population mean. A smaller SE indicates more precision of the sample mean as an estimator of the population mean. The calculation for SE is dependent on the distribution of our data and in the case of our exercise, we calculate it when comparing two means.
Using the formula provided in the step by step solution,\[SE = \sqrt{\frac{{s_{HH}^2}}{{n_{HH}}} + \frac{{s_{SH}^2}}{{n_{SH}}}}\],we replace the sample variances \(s_{HH}^2\) and \(s_{SH}^2\) and the respective sample sizes of Hereford cows (HH) and Brown Swiss/Hereford (SH). By computing the SE, we're quantifying the likelihood of observing the sample means if the null hypothesis were true. It's a crucial step in determining the t-test statistic which informs us if the observed difference in means is significant beyond random chance given the observed variability in our data.
Improvement Advice: To provide additional clarity, examples using actual numbers can be helpful for visual learners. For instance, if \(s_{HH}^2 = 17.8\) and \(s_{SH}^2 = 19.1\) with respective sample sizes of 33 and 51, we would plug those into our formula and calculate a precise SE value. This calculated SE would then be used in subsequent calculations for the t-test statistic.