A female patient has had her red blood cell count tested on 6 occasions. A mean of 4.4 with sample standard deviation, s, of 0.28 was found. Generally, healthy, female adults have a red blood cell count of 4.8. Conduct a hypothesis test to determine if the red blood cell count for this patient is lower than normal. (Use a = 0.05.)
To conduct a hypothesis test to determine if the red blood cell count for the patient is lower than normal, we need to set up the null and alternative hypotheses.
Null Hypothesis (H₀): The red blood cell count for the patient is not lower than normal and is equal to or greater than 4.8.
Alternative Hypothesis (H₁): The red blood cell count for the patient is lower than normal and is less than 4.8.
To test these hypotheses, we will use a one-sample t-test since we have the sample mean and the standard deviation.
The formula to calculate the t-statistic is:
t = (x̄ - μ) / (s / √n)
where:
x̄ is the sample mean,
μ is the population mean,
s is the sample standard deviation,
n is the sample size.
Given:
Sample mean (x̄) = 4.4
Population mean (μ) = 4.8
Sample standard deviation (s) = 0.28
Number of observations (n) = 6
We can now calculate the t-statistic:
t = (4.4 - 4.8) / (0.28 / √6)
t = (-0.4) / (0.28 / 2.449)
t = (-0.4) / (0.114)
t ≈ -3.509
Next, we need to find the critical value for a one-tailed test at a significance level (α) of 0.05. The degrees of freedom (df) is equal to n - 1 = 6 - 1 = 5.
Looking up the critical value from the t-distribution table or using statistical software, we find that the critical value is approximately -2.571.
Since the t-statistic (-3.509) is less than the critical value (-2.571), we have enough evidence to reject the null hypothesis. This means that the red blood cell count for the patient is lower than the normal value of 4.8 at a significance level of 0.05.
Thus, we can conclude that the patient's red blood cell count is statistically significantly lower than normal.