91Ó°ÊÓ

Let \(x\) be a random variable that represents red blood cell (RBC) count in millions of cells per cubic millimeter of whole blood. Then \(x\) has a distribution that is approximately normal. For the population of healthy female adults, the mean of the \(x\) distribution is about 4.8 (based on information from Diagnostic Tests with Nursing Implications, Springhouse Corporation). Suppose that a female patient has taken six laboratory blood tests over the past several months and that the RBC count data sent to the patient's doctor are $$4.9 \quad 4.2 \quad 4.5 \quad 4.1 \quad 4.4 \quad 4.3$$ i. Use a calculator with sample mean and sample standard deviation keys to verify that \(\bar{x}=4.40\) and \(s \approx 0.28\) ii. Do the given data indicate that the population mean RBC count for this patient is lower than \(4.8 ?\) Use \(\alpha=0.05.\)

Step by step solution

01

Compute the Sample Mean

To find the sample mean \( \bar{x} \), add all the RBC count data and divide by the number of observations. Here, the data values are \( 4.9, 4.2, 4.5, 4.1, 4.4, 4.3 \). Calculate as follows: \( \bar{x} = \frac{4.9 + 4.2 + 4.5 + 4.1 + 4.4 + 4.3}{6} = \frac{26.4}{6} = 4.4 \). Thus, \( \bar{x} = 4.4 \).

02

Calculate the Sample Standard Deviation

The sample standard deviation \( s \) is calculated using the formula: \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \), where \( n \) is the number of observations. Compute each deviation \( (x_i - \bar{x}) \), square it, sum all squared deviations, and divide by \( n-1 \). Taking the square root gives \( s \approx 0.28 \). This matches the provided value.

03

Set Up the Hypotheses

We need to determine if the patient's mean RBC count is lower than the population mean. Set up the null hypothesis \( H_0: \mu = 4.8 \) and the alternative hypothesis \( H_a: \mu < 4.8 \). We are conducting a left-tailed test.

04

Compute the Test Statistic

Use the formula for the test statistic \( t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \). Plug in the values: \( \bar{x} = 4.4, \mu = 4.8, s = 0.28, n = 6 \): \( t = \frac{4.4 - 4.8}{\frac{0.28}{\sqrt{6}}} = \frac{-0.4}{0.114} \approx -3.51 \).

05

Determine the Critical Value and Make Decision

For \( \alpha = 0.05 \), find the critical value for a left-tailed test with \( n-1 = 5 \) degrees of freedom using a t-distribution table. The critical value is approximately \(-2.015\). Since \( t = -3.51 \) is less than \(-2.015\), we reject the null hypothesis.

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Key Concepts

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

Sample Mean

The sample mean is a measure of the central tendency of a sample, which provides an average value from a collection of data points. Calculating the sample mean is an integral part of hypothesis testing as it gives us a representative value to compare against the population mean.
To calculate the sample mean, add up all the data points in the sample and divide by the total number of data points.

• For example, if you have RBC counts of 4.9, 4.2, 4.5, 4.1, 4.4, and 4.3, you add these to get 26.4.
• Then divide by the number of observations, which is 6. This gives you a sample mean of 4.4.

Using the sample mean allows us to make comparisons with a known population mean to identify any significant differences.

Sample Standard Deviation

The sample standard deviation measures the amount of variation or dispersion in a set of data values. A small standard deviation indicates that the values are close to the mean, while a large standard deviation indicates that the values are spread out over a wider range.
To calculate the sample standard deviation, follow these steps:

• Subtract the sample mean from each data point and square the result.
• Sum these squared deviations.
• Divide the sum by the number of data points minus one (n-1).
• Lastly, take the square root of this quotient to obtain the standard deviation.

For the RBC count data, this calculation yields a sample standard deviation of approximately 0.28, indicating a modest variability around the mean value.

T-Distribution

The t-distribution is a probability distribution that is symmetrical and bell-shaped, similar to the normal distribution, but it has heavier tails. It is used when the sample size is small and the population standard deviation is unknown, which is often the case in real-world situations.
In hypothesis testing when using small samples, the t-distribution provides a way to estimate the population parameters and make inferences about the population mean.
You utilize the t-distribution when calculating the test statistic for a sample, which helps determine if the sample mean significantly differs from the known population mean.

Critical Value

In hypothesis testing, the critical value is a threshold that the test statistic must exceed in order to reject the null hypothesis. It is determined by the significance level (α) of the test and the degrees of freedom, which depend on the sample size.
For a given confidence level, the critical value helps us decide whether our test statistic is in the "rejection region" of the distribution.

• With a significance level of 0.05 and 5 degrees of freedom in our RBC count test, the critical value from the t-distribution table is approximately -2.015.
• Since the calculated test statistic (-3.51) is less than this critical value, it indicates strong enough evidence to reject the null hypothesis.

Understanding critical values helps in making informed conclusions about the hypothesis being tested.