91Ó°ÊÓ

Let \(x\) be a random variable that represents red blood cell count \((\mathrm{RBC})\) 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 \(\mathrm{RBC}\) count data sent to the patient's doctor are \(\begin{array}{lllllll}4.9 & 4.2 & 4.5 & 4.1 & 4.4 & 4.3\end{array}\) 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 \(\mathrm{RBC}\) count for this patient is lower than \(4.8\) ? Use \(\alpha=0.05\).

Step by step solution

01

Understand the Problem

We need to verify the sample mean \(\bar{x}\) and sample standard deviation \(s\), and then determine if the patient's RBC count is significantly lower than normal, with \(\alpha = 0.05\).

02

Calculate Sample Mean

Calculate the sample mean \(\bar{x}\) by averaging the given RBC values: \( \bar{x} = \frac{4.9 + 4.2 + 4.5 + 4.1 + 4.4 + 4.3}{6} = 4.40 \). This matches the given \(\bar{x}\).

03

Calculate Sample Standard Deviation

Compute the sample standard deviation \(s\) using the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \), where \(x_i\) are the RBC values. Perform calculations to confirm \(s \approx 0.28\).

04

State the Hypotheses

Define the null hypothesis \( H_0: \mu = 4.8 \) (no difference) and the alternative hypothesis \( H_a: \mu < 4.8 \) (mean is lower).

05

Determine the Test Statistic

Calculate the test statistic using the formula \( t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \), where \(\mu = 4.8\), \(\bar{x} = 4.40\), \(s = 0.28\), and \(n = 6\). The computed \(t\) value is \( t = \frac{4.40 - 4.8}{0.28/\sqrt{6}} \approx -3.63\).

06

Find Critical Value and Conclusion

Determine the critical \(t\) value for a one-tailed test with \(df = n-1 = 5\) at \(\alpha = 0.05\). From \(t\)-distribution, \(t_{critical} \approx -2.015\). Since \(-3.63 < -2.015\), reject \(H_0\).

07

State the Conclusion

Since the observed \(t\) value is less than the critical \(t\) value, the data indicates that the patient's mean RBC count is significantly lower than 4.8 at \(\alpha = 0.05\).

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Key Concepts

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

Normal Distribution

The normal distribution is a fundamental concept in statistics, characterized by its bell-shaped curve that is symmetric about the mean. This distribution is often used when analyzing real-world data since many natural phenomena approximately follow it, from test scores to human heights.

• Mean: The average value of the distribution and the center of the bell curve.
• Standard Deviation: Measures the spread or dispersion of values around the mean.

In our exercise, the red blood cell count (RBC) is approximately normally distributed. This allows us to apply techniques like hypothesis testing, which rely on the normality of data. Using normal distribution assumptions helps in making predictions and decisions about population parameters based on sample statistics.

Sample Mean

The sample mean is the average of data points collected from a sample. It's a measure of central tendency, which summarizes the central location of the data. In our exercise:
The sample mean \(\bar{x}\) is calculated by summing all the RBC counts and dividing by the number of samples: \[ \bar{x} = \frac{4.9 + 4.2 + 4.5 + 4.1 + 4.4 + 4.3}{6} = 4.40. \]

• It provides an estimate of the population mean.
• Used as a basis for statistical inference.

Understanding the sample mean is critical when analyzing data, as it gives insights into the general tendency of the dataset and forms the basis for more advanced statistical analysis like hypothesis testing.

Sample Standard Deviation

The sample standard deviation provides a measure of how spread out the values are in your dataset. It is a crucial tool in statistics, helping understand how much variation there is from the mean. Calculating the sample standard deviation involves the following steps:

• Subtract the sample mean from each data point, square the result, and sum them up.
• Divide this sum by the total number of observations minus one.
• Take the square root to get the sample standard deviation.

In the context of the exercise, the standard deviation is approximately 0.28. This value is used to measure the variability in the RBC counts and plays a vital role when calculating the t-distribution for hypothesis testing.

T-Distribution

In hypothesis testing, when the sample size is small or the population variance is unknown, the t-distribution is used instead of the normal distribution. The t-distribution is similar to the normal distribution but has heavier tails, which accommodates more variability when sample sizes are small. In the exercise:
The t-statistic helps us determine whether a sample mean is significantly different from a known or hypothesized population mean. We calculated the test statistic using: \[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]where \(\bar{x} = 4.40\) and \(s = 0.28\). The t-distribution with df = 5 (degrees of freedom) allows us to find the critical value at \(\alpha = 0.05\).
The use of the t-distribution is crucial in our situation to accurately determine statistical significance despite having a small sample, ensuring robust conclusions based on available data.