Math Problem Statement

Use the sample data and confidence level given below to complete parts​ (a) through​ (d).

A drug is used to help prevent blood clots in certain patients. In clinical​ trials, among

44054405

patients treated with the​ drug,

185185

developed the adverse reaction of nausea. Construct a

9595​%

confidence interval for the proportion of adverse reactions.

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Part 1

​a) Find the best point estimate of the population proportion p.

enter your response here

​(Round to three decimal places as​ needed.)

Solution

To find the best point estimate of the population proportion pp, we use the formula:

p^=xn\hat{p} = \frac{x}{n}

where:

  • xx is the number of patients who developed the adverse reaction (nausea),
  • nn is the total number of patients treated.

From the problem:

  • x=185x = 185,
  • n=4405n = 4405.

Now calculate p^\hat{p}:

p^=18544050.042\hat{p} = \frac{185}{4405} \approx 0.042

Thus, the best point estimate of the population proportion pp is:

p^=0.042\hat{p} = 0.042

Would you like further details or explanations on this?

Here are 5 follow-up questions:

  1. How would you calculate the margin of error for a confidence interval?
  2. What formula is used to construct a confidence interval for a population proportion?
  3. How does increasing the confidence level affect the width of the confidence interval?
  4. Why is the point estimate of a population proportion important in statistics?
  5. What assumptions must be met for the confidence interval calculation to be valid?

Tip: Always check if the sample size is large enough to assume a normal distribution for proportions when constructing confidence intervals.

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Math Problem Analysis

Mathematical Concepts

Statistics
Confidence Intervals
Population Proportion

Formulas

Point estimate of population proportion: p̂ = x/n
Confidence interval for population proportion: p̂ ± Z * sqrt[(p̂(1-p̂))/n]

Theorems

Central Limit Theorem (for large sample size approximation)

Suitable Grade Level

Undergraduate/College level