Question

As Americans become more conscious of the importance of good nutrition, some researchers believe that we may be eating less red meat than we used to eat. To test this theory, a researcher selects two groups of 400 subjects each and collects the following sample information on the annual beef consumption now and 10 years ago: $$\begin{array}{lcc}\hline & \text { Ten Years Ago } & \text { This Year } \\\\\hline \text { Sample Mean } & 73 & 63 \\\\\text { Sample Standard Deviation } & 25 & 28 \\\\\hline\end{array}$$ a. The researcher would like to show that per-capita beef consumption has decreased in the last 10 years, so she needs to show that the difference in the averages is greater than 0 . Find a \(99 \%\) lower confidence bound for the difference in the average per-capita beef consumptions for the two groups. b. What conclusions can the researcher draw using the confidence bound from part a?

Short answer

Answer: Yes, there has been a significant decrease in per-capita beef consumption over the past ten years, as the 99% lower confidence bound for the difference in average beef consumption is approximately 8.15 and positive.

Step-by-step solution

Define Variables and Hypotheses

To begin, let's define variables. Let \(X_1\) represent the mean beef consumption from 10 years ago and \(X_2\) represent the mean beef consumption from this year. Our null hypothesis is that there has been no significant change in beef consumption: $$H_0: \mu_1 - \mu_2 = 0$$ Our alternative hypothesis is that per-capita beef consumption has decreased: $$H_1: \mu_1 - \mu_2 > 0$$

Calculate the Lower Confidence Bound

We will now calculate the 99% lower confidence bound for the difference in average beef consumption between 10 years ago and this year. First, we need to determine the confidence interval. Since we are looking for a 99% confidence level, we have \(\alpha = 0.01\). For a one-sided confidence interval, we will use \(z_{\alpha} = 2.33\). Next, let's use the given sample means and standard deviations to calculate the lower confidence bound. Here are the provided values: - \(n_1 = 400\) (Ten Years Ago) - \(\overline{X_1} = 73\) (Ten Years Ago) - \(s_1 = 25\) (Ten Years Ago) - \(n_2 = 400\) (This Year) - \(\overline{X_2} = 63\) (This Year) - \(s_2 = 28\) (This Year) The formula for the lower confidence bound is as follows: $$L = (\overline{X_1} - \overline{X_2}) - z_{\alpha} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$ Plugging in these values: $$L = (73 - 63) - 2.33 \cdot \sqrt{\frac{25^2}{400} + \frac{28^2}{400}} \approx 8.15$$ So, our 99% lower confidence bound for the difference in average beef consumption is approximately \(8.15\).

Draw Conclusions

Since the lower confidence bound for the difference in average beef consumption is positive (approximately \(8.15\)), we can reject the null hypothesis and conclude that there is strong evidence that per-capita beef consumption has decreased in the past 10 years. The researcher can use this information to support the theory that Americans are eating less red meat than they did 10 years ago.

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical method that allows researchers to make inferences about a population based on sample data. The process involves proposing two competing hypotheses: the null hypothesis (ull hypothesis, and the alternative hypothesis (ull hypothesis) and the alternative hypothesis (ull hypothesis). The null hypothesis typically represents a default or no-effect statement, such as no change or no difference in a certain parameter, while the alternative hypothesis suggests the presence of an effect or difference.

To determine which hypothesis is more likely, a test statistic is calculated from the sample data. This statistic is then compared to a distribution, often a z or t-distribution, to calculate the p-value, which reflects the probability of observing the data if the null hypothesis were true. If the p-value is lower than the pre-determined significance level (ull hypothesis), usually set at 0.05 or 0.01, the null hypothesis is rejected in favor of the alternative hypothesis.

It's also important to understand the types of errors that can occur in hypothesis testing: Type I error (false positive) happens if the null hypothesis is wrongly rejected, and Type II error (false negative) occurs if the null hypothesis is incorrectly accepted. The significance level (ull hypothesis) is the probability of making a Type I error.

Standard Deviation

Standard deviation is a measure that quantifies the amount of variability or dispersion in a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (also known as the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

Mathematically, the standard deviation is the square root of the average squared deviations from the mean. The formula for a sample standard deviation (ull hypothesis) is expressed as:
ull hypothesis)

Where ull hypothesis) is the sample size, ull hypothesis) is an individual data point, and ull hypothesis) is the sample mean. Standard deviation is essential for calculating confidence intervals and conducting hypothesis tests as it reflects how much the sample data deviate from the average.

Sample Mean

The sample mean, often denoted by ull hypothesis, represents the average of all the numerical values in a sample. It is calculated by summing up all sample values and then dividing by the number of observations in the sample. Mathematically,
ull hypothesis)

The sample mean serves as an estimate of the population mean and plays a central role in many statistical procedures, including the calculation of confidence intervals and the testing of hypotheses. In the context of the original exercise, the sample mean is used to represent average beef consumption at two different points in time for comparison purposes.

Null Hypothesis

The null hypothesis (ull hypothesis) is a statement used in statistical testing that assumes no significant difference or effect in the case being studied. It represents the theory that any observed difference in the sample data is due to random chance rather than any actual effect or change in the underlying phenomenon.

In our exercise, the null hypothesis posited that there has been no significant change in per-capita beef consumption (ull hypothesis)). This hypothesis provides a baseline against which the alternative hypothesis can be tested. If enough statistical evidence is present to refute the null hypothesis, it can then be rejected in favor of the alternative hypothesis.

Alternative Hypothesis

The alternative hypothesis (ull hypothesis) in statistical analysis proposes that there is a significant difference, effect, or change in the scenario under study. Unlike the null hypothesis, which typically suggests 'no effect' or 'no difference,' the alternative hypothesis is what a researcher aims to support with evidence.

In the context of the original exercise, the alternative hypothesis suggests that the mean beef consumption has decreased over the last ten years (ull hypothesis), which would support the claim of changing dietary habits. The evidence provided by the data that leads to rejecting the null hypothesis in favor of this alternative hypothesis signifies the statistical significance of the observed change.

One-sided Confidence Interval

A one-sided confidence interval provides an estimate of either the lower or upper bound of a parameter based on sample data, indicating that the actual population parameter is either greater than or less than the calculated bound, respectively. For instance, a 99% lower one-sided confidence interval for a mean difference, as calculated in our original exercise, tells us that we can be 99% confident that the true mean difference is greater than the lower bound obtained.

To calculate this interval, we often use the z-score associated with the desired confidence level in one tail of the normal distribution. A z-score represents the number of standard deviations away from the mean a data point is. In the exercise, a z-score for a 99% confidence level (ull hypothesis) was used to calculate the lower confidence bound for the difference in beef consumption. The result provided statistical evidence significant enough to reject the null hypothesis in favor of the alternative.