The doctor estimates that more than 220 of the 500 patients in
his database take a vitamin each day. A random sample of 60
patients shows that 27 of them take a vitamin each day.
Did I miss the question?
yes
To determine whether the proportion of patients who take a vitamin each day is indeed greater than 220 out of 500 or not, we can perform a hypothesis test.
First, let's establish the null and alternative hypotheses:
Null Hypothesis (H0): The proportion of patients who take a vitamin each day is equal to or less than 220/500.
Alternative Hypothesis (H1): The proportion of patients who take a vitamin each day is greater than 220/500.
We can use a one-sample proportion test, specifically a Z-test, to analyze the data. The formula for the test statistic is:
𝑍 = (𝑝̂ - 𝑝0) / √((𝑝0(1-𝑝0))/𝑛)
Where:
- p̂ is the sample proportion (27/60)
- p0 is the hypothesized proportion (220/500)
- n is the sample size (60)
By calculating the test statistic, we can determine the probability of obtaining a sample proportion of 27/60 or greater, assuming the null hypothesis is true.
Next, we need to determine the critical value or the rejection region based on the significance level chosen. Let's assume a significance level of α = 0.05, which is a common choice.
Once we have the test statistic and the critical value, we can compare them to make a decision:
- If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that the proportion of patients who take a vitamin each day is greater than 220/500.
- If the test statistic is less than or equal to the critical value, we fail to reject the null hypothesis and cannot conclude that the proportion is greater.
To perform the calculations and make a final determination, we need the values of p̂, p0, n, the critical value, and the test statistic.