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According to the most recent Bureau of Labor Statistics survey on time use in the United States, the average U.S. man spends \(67.20\) minutes per day eating and drinking. Suppose that a survey of 43 Norwegian men resulted in an average of \(81.10\) minutes per day eating and drinking [Note: This value is consistent with the data in a report by the Organization for Economic Cooperation and Development (Source: http://economix.blogs.nytimes.com/2009/05/05/obesity-and- the-fastness-of-food/)]. Assume that the population standard deviation for all Norwegian men is \(18.30\) minutes. a. Find the \(p\) -value for the test of hypothesis with the alternative hypothesis that the average daily time spent eating and drinking by all Norwegian men is higher than \(67.20\) minutes. What is your conclusion at \(\alpha=.05\) ? b. Test the hypothesis of part a using the critical-value approach. Use \(\alpha=.01\).

Step by step solution

01

State the Hypotheses

First, let's state the null hypothesis \(H_0\) and alternative hypothesis \(H_a\). \n\(H_0: \mu = 67.20\) (The mean (average) time Norwegian men spend eating and drinking per day is equal to the U.S average) \n\(H_a: \mu > 67.20\) (The mean time Norwegian men spend on eating and drinking per day is greater than the U.S. average)

02

Calculate Test Statistic

Now, calculate the z-score (test statistic) using the formula: \n\(z = \frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}\) \nwhere \(\bar{X}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size. \nSubstitute the given values into the formula: \n\(z = \frac{81.10 - 67.20}{\frac{18.30}{\sqrt{43}}}\)

03

Find P-value (part a)

Using a standard normal distribution table or a calculator, find the p-value associated with the calculated z-score. Note that this is an upper-tailed test, so the p-value is the area to the right of the calculated z-score.

04

Compare P-value with Significance Level (part a)

We now compare the p-value with the significance level \(\alpha = 0.05\). If the p-value is less than \(\alpha\), we reject the null hypothesis. If not, we fail to reject the null hypothesis.

05

Find Critical Value (part b)

For part b, we use the critical-value approach. For an alpha level \(\alpha = 0.01\), we find the critical value corresponding to this one-tailed test from the standard normal distribution table.

06

Compare Test Statistic with Critical Value (part b)

The sample data are in the rejection region if the test statistic is greater than the critical value. If so, reject the null hypothesis. If not, fail to reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

The null hypothesis, often denoted as \(H_0\), is a statement assumed to be true unless evidence suggests otherwise. In hypothesis testing, it acts as the foundation of our investigation. Essentially, the null hypothesis is used to verify whether there is "no effect" or no difference in the context of our study.
In the given example, the null hypothesis \(H_0\) states that the average time Norwegian men spend eating and drinking is equal to the U.S. average of 67.20 minutes per day. This presumption implies any observed difference between the two might be due to random chance and not a significant deviation.
When conducting hypothesis testing, the null hypothesis is kept under scrutiny, and the hypothesis test will determine whether it can be rejected based on the data collected.

Alternative Hypothesis

The alternative hypothesis, denoted as \(H_a\), proposes what we suspect or hope to find true. It serves as a counter to the null hypothesis and represents a new perspective or idea about the data. In testing, validating the alternative hypothesis can lead to new insights or conclusions.
For our scenario, the alternative hypothesis \(H_a\) suggests that the average time Norwegian men spend eating and drinking is greater than 67.20 minutes per day. This hypothesis indicates that the Norwegian men, on average, spend more time eating and drinking compared to their U.S. counterparts.

• This hypothesis is particularly useful when we want to see if there's an increase or a specific directional effect.
• In tests like these, we aim to gather enough evidence to support \(H_a\) by rejecting the null hypothesis.

Understanding the alternative hypothesis helps frame the test as we focus on finding evidence to support this claim.

P-Value

The p-value is a crucial component in hypothesis testing, providing a way to measure the evidence against the null hypothesis. It indicates the probability of observing a test statistic as extreme as, or more extreme than, the one obtained from the sample data, assuming that the null hypothesis is true.
In the context of our exercise, once the z-score is calculated, the p-value is derived by finding the area to the right of this z-score on the standard normal distribution. Because we are conducting an upper-tailed test, we focus on this right-tail probability.

• If the p-value is less than the chosen significance level \(\alpha\), often 0.05 or 0.01, it suggests that the null hypothesis is unlikely, hence we reject it.
• The smaller the p-value, the stronger the evidence against the null hypothesis, suggesting the alternative hypothesis might be true.

The p-value helps determine whether the observed data is consistent with the null hypothesis or if there is sufficient evidence to support the alternative theory.

Critical-Value Approach

The critical-value approach is another technique to conduct hypothesis testing and involves comparing the test statistic against a threshold value, known as the critical value. This approach is grounded on the decision rule principle, where rejection or acceptance of the null is contingent upon whether the test statistic falls within a specific region of the distribution known as the rejection region.
For the exercise, given a significance level \(\alpha = 0.01\), we find the critical value corresponding to this one-tailed test from the standard normal distribution table. The threshold is determined by the point beyond which only 1% of the data would occur if the null hypothesis is true.

• If the calculated test statistic exceeds the critical value, it implies the test statistic falls in the rejection region, leading us to reject the null hypothesis.
• Conversely, if the test statistic does not exceed the critical value, we fail to reject the null hypothesis.

Utilizing the critical-value approach provides a clear-cut decision-making process by defining a fixed point in the distribution to judge the statistical significance of the results.