Problem 29 - Basic Statistics for Business and Economics with Student

Question

A new weight-watching company, Weight Reducers International, advertises that those who join will lose, on the average, 10 pounds the first two weeks with a standard deviation of 2.8 pounds. A random sample of 50 people who joined the new weight reduction program revealed the mean loss to be 9 pounds. At the .05 level of significance, can we conclude that those joining Weight Reducers on average will lose less than 10 pounds? Determine the \(p\) -value.

Step-by-Step Solution

Answer

Reject the null hypothesis; the mean loss is less than 10 pounds (\(p\text{-value} = 0.0057\)).

Step 1: State the Hypotheses

We begin by stating the null and alternative hypotheses. The null hypothesis (\(H_0\)) is that the average weight loss is 10 pounds, i.e., \( \mu = 10 \). The alternative hypothesis (\(H_a\)) is that the average weight loss is less than 10 pounds, i.e., \( \mu < 10 \).

Step 2: Determine the Test Statistic

Since the sample size is 50, which is greater than 30, we can use a Z-test for the hypothesis testing. The formula for the test statistic is given by: \[ Z = \frac{{\bar{x} - \mu}}{{\sigma/\sqrt{n}}} \] Where: \( \bar{x} = 9 \) (sample mean), \( \mu = 10 \) (population mean under \(H_0\)), \( \sigma = 2.8 \) (population standard deviation), and \( n = 50 \) (sample size).

Step 3: Calculate the Test Statistic

Substitute the given values into the formula: \[ Z = \frac{{9 - 10}}{{2.8/\sqrt{50}}} = \frac{{-1}}{{2.8/7.071}} = \frac{{-1}}{{0.396}} \approx -2.53 \] Thus, the test statistic is approximately \(-2.53\).

Step 4: Find the p-value

The \(p\)-value is found by looking up the calculated Z value (-2.53) in the standard normal distribution table. The \(p\)-value associated with \(Z = -2.53\) is approximately 0.0057.

Step 5: Make a Decision

Compare the \(p\)-value to the significance level (\(\alpha = 0.05\)). Since the \(p\)-value (0.0057) is less than \(\alpha\), we reject the null hypothesis \(H_0\).

Step 6: Conclusion

At the 0.05 level of significance, there is enough evidence to conclude that those joining Weight Reducers on average lose less than 10 pounds in the first two weeks.

Key Concepts

Understanding the Z-Test

The Z-test is a statistical method used to determine if there's a significant difference between sample data and the population mean. It's commonly used when the sample size is large (usually over 30) because it assumes a normal distribution.
For this particular problem, the Z-test helps us assess whether the weight loss observed in our sample is significantly different from the advertised 10 pounds.

• We assess the sample mean compared to the population mean.
• Population standard deviation is known, which is crucial for using a Z-test.
• The "Z" refers to the Z-score, which is the number of standard deviations an element is from the mean.

This is why we use the formula: \[Z = \frac{{\bar{x} - \mu}}{{\sigma/\sqrt{n}}}\]Plug the values into the formula to find your specific Z-score.
Understanding Z-tests helps determine if our sample deviates from expectations or norms.

Decoding the P-Value

P-value, in hypothesis testing, helps decide whether to reject the null hypothesis. It tells us the probability of observing the results when the null hypothesis is true.
In simpler terms, a small p-value indicates strong evidence against the null hypothesis, meaning our sample result was not due to random chance.

• P-values range from 0 to 1.
• A small p-value (typically ≤ 0.05) suggests rejecting the null hypothesis.
• In our exercise, a p-value of 0.0057 is quite small, hinting that the sample mean of 9 pounds is not a product of chance alone.

By looking at p-values, you can objectively test assumptions and theory, which in our case is about weight loss consistency.

Grasping the Null Hypothesis

The null hypothesis (\(H_0\)) is a statement used in statistics that assumes no effect or no difference. It's our starting point in hypothesis testing, representing the concept that there is no change or surprise in the data.
In our weight-loss scenario:

• The null hypothesis claims that average weight loss is indeed 10 pounds.
• It's what you assume to be true initially and seek evidence against.
• Rejecting the null hypothesis implies finding sufficient data to believe an alternative narrative.

Null hypotheses are the benchmark or status quo in statistical tests, guiding initial expectations.

Clarifying the Alternative Hypothesis

The alternative hypothesis (\(H_a\)) is the statement you want to prove when conducting an experiment. It directly contradicts the null hypothesis, suggesting an effect or difference exists.
In the context of our weight-loss exercise:

• The alternative hypothesis posits that the average weight loss is less than 10 pounds.
• It suggests a change or deviation from what's claimed.
• A successful statistical test will find enough evidence to support the alternative hypothesis.

The role of the alternative hypothesis is to set up what you are testing for and to help clarify your research or testing objectives.