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Chapter 9: Problem 56

Pulse and Gender: Cl Using data from NHANES, we looked at the pulse rate for nearly 800 people to see whether it is plausible that men and women have the same population mean. NHANES data are random and independent. Minitab output follows.

Short Answer

Expert verified
The short answer would depend on the actual values obtained from the Minitab output. However, the decision to reject or retain the null hypothesis will be made based on the P-value and the cut-off for significance (typically 0.05). For instance, if the P-value is found to be less than 0.05, then the conclusion will be that there is a significant difference in mean pulse rates between men and women. Conversely, if the P-value is larger than 0.05, then the conclusion will be that there is no significant difference in mean pulse rates between men and women.

Step by step solution

01

Formulate the Hypotheses

Start by setting up the null hypothesis \(H_0\), which would posit that the population mean pulse rate is the same for both men and women, and the alternative hypothesis \(H_1\), which would suggest that there is a significant difference in the mean pulse rates of men and women. Essentially, the hypotheses would be formulated as: \(H_0: \mu_{men} = \mu_{women}\) , \(H_1: \mu_{men} \neq \mu_{women}\).
02

Conduct the Statistical Test

Use the Minitab output to conduct the statistical test. This output usually includes the sample means, standard deviations, and sample sizes for both groups as well as the t-statistic and P-value for the difference of means. The t-statistic provides an estimate of the magnitude of the difference in means relative to the variability within groups, while the P-value measures the strength of evidence against the null hypothesis.
03

Interpret the Results

Decide whether to reject the null hypothesis based on the P-value. A small P-value (typically less than 0.05) provides strong evidence against the null hypothesis, suggesting that you should reject it and conclude that there is a significant difference in mean pulse rates between men and women. Conversely, a large P-value indicates weak evidence against the null hypothesis, which suggests that you do not reject it, implying that there is no significant difference in mean pulse rates between the genders.

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis, often denoted as \( H_0 \), represents a statement of no effect or no difference. It serves as the default or starting assumption. For the pulse rate study, the null hypothesis proposes that there is no difference in the average pulse rates between men and women. Mathematically, it is expressed as:
  • \( H_0: \mu_{men} = \mu_{women} \)
The null hypothesis assumes that any observed difference in sample means is due to random chance rather than a true difference in the population. It's essential because it provides a baseline against which we can measure the strength of evidence in favor of an effect.
Alternative Hypothesis
The alternative hypothesis, denoted as \( H_1 \) or \( H_a \), is what you want to prove. It suggests that there is a statistically significant effect or difference. In the pulse rate analysis, the alternative hypothesis indicates that there is a difference in the mean pulse rates of men and women:
  • \( H_1: \mu_{men} eq \mu_{women} \)
This hypothesis challenges the status quo proposed by the null hypothesis. It is usually what researchers set out to demonstrate. When interpreting results, a key goal is to determine whether data provide sufficient evidence to support the alternative hypothesis.
T-Statistic
The t-statistic is a ratio that compares the difference between the sample means to the variability in the samples. It is crucial in determining whether the observed differences could have happened by random chance.
  • Formula: \( t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \)
Where \( \bar{x}_1 \) and \( \bar{x}_2 \) are the sample means, \( s_p \) is the pooled standard deviation, and \( n_1 \) and \( n_2 \) are the sample sizes.
The larger the t-statistic, the more evidence we have against the null hypothesis. Essentially, it shows how many standard deviations the sample difference is away from zero, the hypothesized population difference under the null.
P-Value
The P-value measures the probability of obtaining results at least as extreme as the observed ones, assuming the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis.
  • Common threshold: \( \alpha = 0.05 \)
If the P-value is less than the significance level (usually 0.05), we reject the null hypothesis. This suggests that the observed differences are unlikely to be due to chance alone.
Conversely, a large P-value suggests that we do not have enough evidence to reject the null, meaning the difference in pulse rates is not statistically significant. P-values help in making informed decisions about the validity of the null hypothesis.

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Most popular questions from this chapter

Confidence Interval Changes State whether each of the following changes would make a confidence interval wider or narrower. (Assume that nothing else changes.) a. Changing from a \(95 \%\) level of confidence to a \(90 \%\) level of confidence b. Changing from a sample size of 30 to a sample size of 20 c. Changing from a standard deviation of 3 inches to a standard deviation of \(2.5\) inches

Construct two sets of body temperatures (in degrees Fahrenheit, such as \(96.2{ }^{\circ} \mathrm{F}\) ), one for men and one for women, such that the sample means are different but the hypothesis test shows the population means are not different. Each set should have three numbers in it.

Presidents' Ages at Inauguration A \(95 \%\) confidence interval for the ages of the first six presidents at their inaugurations is \((56.2,59.5) .\) Either interpret the interval or explain why it should not be interpreted.

Student Ages The mean age of all 2550 students at a small college is \(22.8\) years with a standard deviation is \(3.2\) years, and the distribution is right- skewed. A random sample of 4 students' ages is obtained, and the mean is \(23.2\) with a standard deviation of \(2.4\) years. a. \(\mu=? \quad \sigma=? \quad \bar{x}=? \quad s=?\) b. Is \(\mu\) a parameter or a statistic? \(c\). Are the conditions for using the CLT fulfilled? What would be the shape of the approximate sampling distribution of many means, cach from a sample of 4 students? Would the shape be right-skewed, Normal, or left-skewed?

Women's Heights Assume women's heights are approximately Normally distributed with a mean of 65 inches and a standard deviation of \(2.5\) inches. Which of the following questions can be answered using the Central Limit Theorem for sample means as needed? If the question can be answered, do so. If the question cannot be answered, explain why the Central Limit Theorem cannot be applied. a. Find the probability that a randomly selected woman is less than 63 inches tall. b. If five women are randomly selected, find the probability that the mean height of the sample is less than 63 inches. c. If 30 women are randomly selected, find the probability that the mean height of the sample is less than 63 inches.
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