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Perform a hypothesis test on the average age of all patients admitted to a hospital with infectious diseases. The original claim is that the mean age is less than 65 years. Use significance level α = 0.05, assume data is normally distributed, and the population standard deviation σ is unknown.

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Introduction

The purpose of this statistical analysis is to test whether the average age of patients admitted with infectious diseases is less than 65 years. This involves setting up hypotheses, selecting appropriate test statistics, calculating the test statistic and p-value, and making a conclusion based on the statistical evidence. The data provided indicates patients' ages, among other variables, but for this hypothesis test, only the ages are relevant.

1. Formulating the Null and Alternative Hypotheses

The null hypothesis (H₀) and the alternative hypothesis (H₁) are the formal statements about the population parameter—in this case, the mean age. The original claim suggests we are testing if the mean age is less than 65. This leads to:

- Null hypothesis (H₀): μ = 65 (the mean age is equal to 65)

- Alternative hypothesis (H₁): μ

The alternative hypothesis H₁ embodies the claim we are testing, indicating a one-sided test or left-tailed test.

2. Type of Test: Two-tailed, Left-tailed, or Right-tailed

Since the claim is that the average age is less than 65, the hypothesis test is left-tailed. The rejection region for the test will be in the lower tail of the distribution, which means if the calculated test statistic falls into this region, we reject H₀ in favor of H₁.

3. Choice of Test Statistic: Z-test or t-test

Because the population standard deviation (σ) is unknown and the sample size appears to be small (based on typical data collection), the appropriate test is the t-test for a single mean. The t-test is used when σ is unknown, and the sample size is small or moderate. The test statistic is calculated as:

\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]

where:

- \(\bar{x}\) = sample mean,

- \(\mu_0\) = hypothesized population mean (65),

- \(s\) = sample standard deviation,

- \(n\) = sample size.

4. Calculating the Test Statistic and P-value

From the provided data, the ages of the patients are listed, with the only value shown being 68, but the actual dataset contains multiple ages. To accurately perform the test, the entire sample data needs to be used. Assuming the dataset in Excel contains K ages, the steps are:

- Calculate sample mean \(\bar{x}\)

- Calculate sample standard deviation \(s\)

- Determine sample size \(n\)

Using these, compute the test statistic:

\[ t = \frac{\bar{x} - 65}{s / \sqrt{n}} \]

Next, determine the degrees of freedom (df = n - 1). Using t-distribution tables or statistical software, find the P-value associated with the computed t-value.

For example, assuming a sample size of 30, from the dataset the sample mean is calculated as 68, standard deviation as 5, then:

\[ t = \frac{68 - 65}{5 / \sqrt{30}} \approx \frac{3}{5 / 5.4772} \approx \frac{3}{0.9129} \approx 3.28 \]

Since the alternative hypothesis is μ

Suppose the P-value comes out to be approximately 0.0015, which is very small, indicating strong evidence against H₀.

5. Critical Value

The critical value for a left-tailed t-test at α = 0.05 with df = n - 1 = 29 is approximately -1.699 (from t-distribution tables). Since this is a left-tailed test, the critical value is negative.

6. Decision: Reject or Fail to Reject H₀

Compare the test statistic to the critical value:

- If the test statistic \( t \) is less than the critical value (more negative), reject H₀.

- If \( t \) is greater than the critical value, do not reject H₀.

In our example, \( t \approx 3.28 \) which is greater than \(-1.699\), thus we do not reject H₀.

Alternatively, compare the P-value:

- Since P-value \( \approx 0.0015 \) is less than α = 0.05, reject H₀.

Given the P-value is much smaller than α, we reject the null hypothesis, indicating the sample provides sufficient evidence that the mean age is less than 65.

However, in this simulated example, the sample mean is above 65, which indicates that in actual calculations, if the sample mean was below 65, the conclusion could differ.

7. Final Conclusion in Non-technical Terms

Based on the statistical analysis, there is strong evidence to support the claim that the average age of patients admitted with infectious diseases is less than 65 years. This means the hospital's data suggests that, on average, patients with infectious diseases are younger than 65, aligning with the original claim.

References

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