According to a national study, 45% of american adults have hypertension, 20% have hypotension, and 35% have normal blood pressure. A sample of 150 adults was taken and the following data were collected. Hypertensive: 54, Normal: 69, Hypotensive: 27. Test at the 5% level whether the distribution of blood pressure categories differs from the national numbers.

To test whether the distribution of blood pressure categories differs from the national numbers, we can use a chi-square test of independence. This test is appropriate when you have categorical data and want to determine if there is a relationship between two variables.

In this case, we have two categorical variables: the blood pressure categories (hypertensive, normal, and hypotensive) and the national numbers (45% hypertension, 20% hypotension, and 35% normal).

To conduct the chi-square test, we need to set up a hypothesis:

Null Hypothesis (H0): The distribution of blood pressure categories in the sample is the same as the national numbers.
Alternative Hypothesis (Ha): The distribution of blood pressure categories in the sample is different from the national numbers.

Next, we calculate the expected frequencies for each category assuming that the null hypothesis is true. We can do this by multiplying each national percentage by the total sample size of 150:

Expected hypertensive count: 150 * 0.45 = 67.5
Expected normal count: 150 * 0.35 = 52.5
Expected hypotensive count: 150 * 0.20 = 30

Now we can set up a contingency table with the observed and expected frequencies:

Hypertensive Normal Hypotensive
Observed count 54 69 27
Expected count 67.5 52.5 30

To compute the chi-square test statistic, we use the formula:

χ^2 = Σ((Observed - Expected)^2 / Expected)

Calculating the chi-square statistic for this data:

χ^2 = ((54 - 67.5)^2 / 67.5) + ((69 - 52.5)^2 / 52.5) + ((27 - 30)^2 / 30)

Next, we find the degrees of freedom for the test, which is given by the formula:

df = (number of rows - 1) * (number of columns - 1)

In this case, df = (3 - 1) * (3 - 1) = 2 * 2 = 4.

To determine whether the calculated chi-square statistic is statistically significant, we compare it to the critical value from the chi-square distribution table at a significance level of 0.05 and degrees of freedom of 4.

If the calculated chi-square statistic is greater than the critical value, we reject the null hypothesis, indicating that the distribution of blood pressure categories differs from the national numbers. Otherwise, we fail to reject the null hypothesis.

Alternatively, we can calculate the p-value associated with the calculated chi-square statistic. If the p-value is less than 0.05 (the significance level), we reject the null hypothesis.

By following these steps, we can determine whether the distribution of blood pressure categories in the sample differs from the national numbers at a significance level of 0.05.