Problem 26 - Calculus for Biology and Medicine

Question

To test a new drug for lowering cholesterol, 72 people with elevated cholesterol receive the drug; 51 of them show reduced cholesterol levels. Estimate the probability that the drug lowers cholesterol, and construct a \(95 \%\) confidence interval.

Step-by-Step Solution

Answer

The probability estimate is 0.7083, with a 95% confidence interval of (0.6038, 0.8128).

Step 1: Define the Problem

We are asked to estimate the probability that the drug lowers cholesterol and to construct a 95% confidence interval for this probability. The observed data is that out of 72 people, 51 showed reduced cholesterol levels.

Step 2: Calculate the Sample Proportion

The sample proportion \( \hat{p} \) is given by the number of successes (people who showed reduced cholesterol) divided by the total number of trials (people who took the drug). Thus, \( \hat{p} = \frac{51}{72} \approx 0.7083 \).

Step 3: Use the Standard Error Formula

The standard error (SE) of the sample proportion is calculated using the formula: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]Substitute \( \hat{p} = 0.7083 \) and \( n = 72 \):\[ SE = \sqrt{\frac{0.7083 \times (1 - 0.7083)}{72}} \approx 0.0536 \]

Step 4: Find the Critical Value for 95% Confidence

For a 95% confidence interval, we use a critical value from the standard normal distribution (Z-distribution), which is approximately 1.96.

Step 5: Construct the Confidence Interval

The 95% confidence interval for the population proportion \( p \) is given by:\[ \hat{p} \pm Z \times SE \]Substitute \( \hat{p} = 0.7083 \), \( Z = 1.96 \), and \( SE = 0.0536 \):\[ 0.7083 \pm 1.96 \times 0.0536 \approx (0.6038, 0.8128) \]

Step 6: Interpret the Results

The estimated probability that the drug lowers cholesterol is approximately 0.7083. The 95% confidence interval suggests that we are 95% confident that the true probability that the drug lowers cholesterol is between 0.6038 and 0.8128.

Key Concepts

Sample Proportion

The sample proportion is a fundamental concept in statistics when we want to infer something about a population based on a sample. In the original exercise, we had 72 people who took the drug, and out of these, 51 showed reduced cholesterol levels. This is our observed data, and from it, we can calculate the sample proportion. To find the sample proportion, divide the number of successes (in this case, the people showing reduced cholesterol) by the total number of trials (the total number of people in the study). So, if you take the number 51 and divide it by 72, you get approximately 0.7083. This means about 70.83% of the sample showed a reduction in cholesterol. The sample proportion, often denoted by the symbol \( \hat{p} \), is a key figure because it serves as an estimate of the unknown true proportion of the population. Understanding sample proportion helps us make predictions or statements about the entire population that was not directly observed.

Standard Error

The standard error helps us understand how much variability might exist in the sample proportion compared to the true population proportion. It provides an estimate of the standard deviation of the sampling distribution of the sample proportion.The formula to calculate standard error (SE) of the sample proportion is: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]Here, \( \hat{p} \) is the sample proportion (which is 0.7083 in our exercise), and \( n \) is the sample size, which is 72.Substituting the values into the formula gives: \[ SE = \sqrt{\frac{0.7083 \times (1 - 0.7083)}{72}} \approx 0.0536 \]This result shows how much \( \hat{p} \) is expected to vary from the true population proportion with repeated sampling. A smaller standard error indicates that the sample proportion is likely to be a good estimate of the population proportion. It is crucial in creating confidence intervals, which helps us understand the range in which we expect the true population parameter to fall.

Critical Value

The critical value is an essential part of constructing confidence intervals, as it helps determine how wide the interval will be. It comes from the standard normal distribution and varies based on the confidence level you want. For a 95% confidence interval, the critical value is generally 1.96. This value implies that if you were to take many samples and construct a confidence interval from each of them, about 95% of those intervals would contain the true population parameter.To use the critical value for constructing the confidence interval, you multiply it by the standard error (SE). In this context:\[ CI = \hat{p} \pm Z \times SE \]Where:- \( \hat{p} \) is the sample proportion.- \( Z \) is the critical value of 1.96.- \( SE \) is the standard error, calculated previously as 0.0536.So, the confidence interval becomes:\[ 0.7083 \pm 1.96 \times 0.0536 \approx (0.6038, 0.8128) \]This interval tells us that we can be 95% sure that the proportion of the entire population whose cholesterol is reduced by the drug is between 60.38% and 81.28%.