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Chapter 19: Problem 39

In a 2013 study, researchers compared various measurements on overweight firstborn and second-born middle-aged men. \({ }^{7}\) They found that first- borns had a significantly higher weight \((P=0.013\) ) than second-borns, but no significant difference in total cholesterol \((P=0.74)\). Explain carefully why \(P=0.013\) means there is evidence that first-born middle-aged men may have higher weights than second-borns and why \(P=0.74\) provides no evidence that first-born middle-aged men may have different total cholesterol levels than second-borns.

Short Answer

Expert verified
The P-value of 0.013 provides evidence of higher weight in firstborns, while 0.74 offers no evidence for cholesterol differences.

Step by step solution

01

Understanding P-Values

A P-value measures the probability of obtaining results at least as extreme as the observed results, assuming that the null hypothesis is true. It helps to determine the statistical significance of the observed effect.
02

Interpreting Lower P-Value for Weight Comparison

A P-value of 0.013 is considered statistically significant because it is below the usual threshold of 0.05. This indicates that there is only a 1.3% probability that the observed higher weight in firstborns compared to second-borns could be due to chance, providing strong evidence against the null hypothesis.
03

Interpreting Higher P-Value for Cholesterol Comparison

A P-value of 0.74 is not statistically significant as it is well above the threshold of 0.05. This means there is a 74% probability that any difference in cholesterol levels between firstborns and second-borns could just be due to random variation, thus not providing sufficient evidence 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.

P-value
In statistics, the P-value is a crucial concept. It helps us understand the strength of the evidence against the null hypothesis. A P-value is essentially the probability of obtaining results that are as extreme as those observed, given that the null hypothesis is true.
For instance, when comparing the weights of firstborn and second-born middle-aged men, the observed P-value was 0.013. This value is below the typical significance threshold of 0.05, suggesting that there's only a 1.3% chance the observed difference happened by random chance if there were actually no difference in reality.
This low probability provides strong evidence against the null hypothesis, implying a real difference in weights. On the flip side, a P-value of 0.74 for cholesterol levels suggests a 74% chance of the observed difference due to random variation. It implies no evidence against the null hypothesis and hence no significant difference.
Null Hypothesis
The null hypothesis is a cornerstone of statistical analysis. It is a statement of no effect or no difference, serving as a starting point for testing. In many tests, the null hypothesis might state that two groups, such as firstborns and second-borns, are alike in terms of a specific characteristic.
For example, when assessing weight or cholesterol in the study, the null hypothesis would assert there's no difference between firstborn and second-born men.
We use statistical tests to determine whether we can reject this hypothesis. If the evidence, as indicated by a low P-value, contradicts the null hypothesis, we may conclude there is a statistically significant difference between the groups.
Statistical Analysis
Statistical analysis is the method by which data is collected, reviewed, and interpreted. It allows scientists to draw meaningful conclusions from data sets. This involves calculating P-values, comparing them to significance thresholds, and making inferences about hypotheses.
In our study example, various statistical analyses were performed to compare the weights and cholesterol levels of firstborn and second-born middle-aged men. The analysis revealed that the weight difference was statistically significant (P=0.013), while the cholesterol difference was not (P=0.74).
This kind of analysis helps to clarify which differences are likely real and which may not be, enabling evidence-based conclusions in research.

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Most popular questions from this chapter

Byron claims that the probability Ohio State and Alabama will play in the national championship football game this year is \(33 \%\). The number \(33 \%\) is (a) the proportion of times Ohio State and Alabama have played in the championship game in the past. (b) Byron's personal probability that Ohio State and Alabama will play in the championship football game this year. (c) the area under a Normal density curve. (d) all of the above.

When our brains store information, complicated chemical changes take place. In trying to understand these changes, researchers blocked some processes in brain cells taken from rats and compared these cells with a control group of normal cells. They say that "no differences were seen" between the two groups in four response variables. They give \(P\)-values \(0.45,0.83,0.26\), and \(0.84\) for these four comparisons. \({ }^{6}\) Which of the following statements is correct? (a) It is literally true that "no differences were seen." That is, the mean responses were exactly alike in the two groups. (b) The mean responses were exactly alike in the two groups for at least one of the four response variables measured, but not for all of them. (c) The statement "no differences were seen" means that the observed differences were not statistically significant at the significance level used by the researchers. (d) The statement "no differences were seen" means that the observed differences were all less than 1 (and were actually \(0.45,0.83,0.26\), and \(0.84\) for these four comparisons).

Americans spend more than \(\$ 30\) billion annually on a variety of weight loss products and services. In a study of retention rates of those using the Rewards Program at Jenny Craig in 2005, it was found that about \(18 \%\) of those who began the program dropped out in the first four weeks. \({ }^{14}\) Assume we have a random sample of 300 people beginning the program. (a) What is the mean number of people who would drop out of the Rewards Program within four weeks in a sample of this size? What is the standard deviation? (b) What is the probability that at least 235 people in the sample will still be in the Rewards Program after the first four weeks? Check that the Normal approximation is permissible and use it to find this probability. If your software allows, find the exact binomial probability and compare the two results.

Alysha makes \(40 \%\) of her free throws. She takes five free throws in a game. If the shots are independent of each other, the probability that she misses the first two shots but makes the other three is about (a) \(0.230\). (b) \(0.115\). (c) \(0.023\). (d) \(0.600\).

According to the National Survey of Student Engagement (NSSE), the average amount of time that first-year college students spent preparing for class (studying, reading, writing, doing homework or lab work, analyzing data, rehearsing, and other academic activities) in 2013 was \(14.25\) hours per week. Your college wonders if the average \(\mu\) for its first-year students in 2013 differed from the national average. A random sample of 500 students who were first-year students in 2013 claims to have spent an average of \(x^{-} \bar{x}=13.4\) hours per week on homework in their first year. What are the null and alternative hypotheses for a comparison of first-year students at your college with national first-year students in 2013 ? (a) \(H_{0}: \mathrm{x}^{-} \bar{x}=14.25, H_{a}: \mathrm{x}^{-} \bar{x} \neq 14.25\) (b) \(H_{0}: x^{-} \bar{x}=13.4, H_{a}: x^{-} \bar{x}>13.4\) (c) \(H_{0}: \mu=14.25, H_{a}: \mu \neq 14.25\) (d) \(H_{0}: \mu=13.4, H_{a}: \mu>13.4\)
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