91Ó°ÊÓ

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. What is the value of the sample test statistic? (c) Find (or estimate) the \(P\) -value. Sketch the sampling distribution and show the area corresponding to the \(P\) -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \(\alpha\) ? (e) State your conclusion in the context of the application. Total blood volume (in ml) per body weight (in \(\mathrm{kg}\) ) is important in medical research. For healthy adults, the red blood cell volume mean is about \(\mu=28 \mathrm{ml} / \mathrm{kg}\) (Reference: Laboratory and Diagnostic Tests, F. Fischbach). Red blood cell volume that is too low or too high can indicate a medical problem (see reference). Suppose that Roger has had seven blood tests, and the red blood cell volumes were \(\begin{array}{lllllll}32 & 25 & 41 & 35 & 30 & 37 & 29\end{array}\) The sample mean is \(\bar{x} \approx 32.7 \mathrm{ml} / \mathrm{kg}\). Let \(x\) be a random variable that represents Roger's red blood cell volume. Assume that \(x\) has a normal distribution and \(\sigma=4.75 .\) Do the data indicate that Roger's red blood cell volume is different (either way) from \(\mu=28 \mathrm{ml} / \mathrm{kg}\) ? Use a \(0.01\) level of significance.

Step by step solution

01

State the Hypotheses and Significance Level

The level of significance is given as \( \alpha = 0.01 \). This means that there is a 1% risk of rejecting the null hypothesis when it is actually true. The null hypothesis \((H_0)\) is that Roger's red blood cell volume is equal to the population mean, \( \mu = 28 \ \mathrm{ml/kg} \). The alternate hypothesis \((H_a)\) is that Roger's red blood cell volume is different from 28, which can be expressed as \( H_a: \mu eq 28 \ \mathrm{ml/kg} \). Since we are testing if the volume is different from the mean, this is a two-tailed test.

02

Select the Sampling Distribution

Since the population standard deviation \( \sigma \) is known and the sample size is small (n=7), we will use the normal distribution for the sampling distribution of the test statistic. The test statistic for a known \( \sigma \) is calculated as \( z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \). Substituting the given values, we have \( \bar{x} = 32.7 \), \( \mu = 28 \), \( \sigma = 4.75 \), and \( n = 7 \). Hence the test statistic calculated is \( z = \frac{32.7 - 28}{4.75/\sqrt{7}} \approx 3.48 \).

03

Determine the P-Value and Sketch Distribution

To find the \( P \)-value corresponding to the calculated test statistic \( z \approx 3.48 \), we look for the probability in the standard normal distribution table. The \( P \)-value is the area in both tails beyond \( |z| = 3.48 \). The \( P \)-value for \( z = 3.48 \) is approximately \( 2 \times 0.00025 = 0.0005 \). The distribution sketch shows the normal curve with \( z = 3.48 \) in both tails, indicating the areas for extreme values.

04

Make a Decision Regarding the Null Hypothesis

Since the \( P \)-value (0.0005) is less than the significance level \( \alpha = 0.01 \), we reject the null hypothesis \( H_0 \). This means that the test provides sufficient evidence to conclude that Roger's red blood cell volume is statistically different from the population mean at a 1% significance level.

05

State Conclusions in Context

Given the rejection of the null hypothesis, we conclude that there is significant evidence that Roger's red blood cell volume is different from the average healthy adult measurement of 28 ml/kg. Therefore, it indicates a potential medical condition worth further investigation.

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Key Concepts

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

Level of Significance

In hypothesis testing, the level of significance, denoted as \( \alpha \), plays a crucial role in decision making. It represents the probability of rejecting the null hypothesis when it is actually true, often termed as a Type I error. In this exercise, the level of significance is stated as \( \alpha = 0.01 \). This means there is a 1% risk of making a Type I error.

Choosing a level of significance depends on the context of the study. A lower \( \alpha \) indicates a stricter criterion for rejection, which is especially useful in scenarios where consequences of a Type I error are severe, like in medical research. Here, our goal is to determine if Roger's red blood cell volume significantly differs from the established mean, making the 1% threshold both prudent and necessary.

Null and Alternate Hypotheses

Null and alternate hypotheses form the cornerstone of hypothesis testing. The null hypothesis \((H_0)\) generally states that there is no effect or difference. It is the hypothesis that researchers aim to test against. In this scenario, the null hypothesis is that Roger's red blood cell volume is equal to the population mean of 28 ml/kg, detailed as \(H_0: \, \mu = 28 \ \mathrm{ml/kg}\).

Contrarily, the alternate hypothesis \((H_a)\) is the statement that indicates a difference or effect exists. For our exercise, \(H_a: \, \mu eq 28 \ \mathrm{ml/kg}\) suggests that Roger's red blood cell volume is not equal to the population mean, implying a potential medical anomaly. Together, these hypotheses set up a framework to test our assumptions.

Two-Tailed Test

A two-tailed test is employed when researchers are interested in determining whether there is a significant difference in either direction from a specific value. In Roger's case, we want to test if his red blood cell volume is either higher or lower than the population mean of 28 ml/kg.

This type of testing is characterized by having critical regions in both tails of the distribution curve, allowing for the detection of deviations in either direction. Since we are checking for any potential deviation from the mean, a two-tailed test is appropriate. It is more stringent as it tests against an area in both extremes, suggesting thoroughness in concluding whether the data behaves unusually compared to the norm.

Sampling Distribution

The sampling distribution is a fundamental concept in hypothesis testing, as it helps determine the behavior of the test statistic under the null hypothesis. Here, since the population standard deviation \( \sigma \) is known, the normal distribution is suitable. This is true even though we have a small sample size \((n=7)\) because we are assuming the data is normally distributed.

The test statistic is calculated using the formula:

• \[ z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \]

Given \( \bar{x} \approx 32.7 \), \( \mu = 28 \), \( \sigma = 4.75 \), and \( n = 7 \), the computation results in \( z \approx 3.48 \). This key value enables the determination of the \( P \)-value, which in turn guides the decision making process regarding the null hypothesis.