Math Problem Statement

the sample proportion \hat{\pi} for a random sample of 60 US adults regarding whether they have not read a book in any format in the past 12 months is approximately normally distributed with a mean of 0.23 and a standard deviation of approximately 0.0543.

Solution

The problem describes the sample proportion (π^\hat{\pi}) from a random sample of 60 U.S. adults regarding whether they haven't read a book in the past 12 months. The given values are:

  • Mean (μπ^\mu_{\hat{\pi}}) = 0.23
  • Standard deviation (σπ^\sigma_{\hat{\pi}}) = 0.0543
  • Sample size n=60n = 60

Since the sample proportion π^\hat{\pi} is approximately normally distributed, we can use this information to answer several questions or compute probabilities.

Would you like help with calculating probabilities, constructing confidence intervals, or testing hypotheses using this information?

Here are 5 potential questions to explore:

  1. What is the probability that the sample proportion is less than 0.2?
  2. What is the probability that the sample proportion is more than 0.25?
  3. How do you construct a 95% confidence interval for the true population proportion?
  4. How do you test if the population proportion is significantly different from 0.20?
  5. What is the probability that the sample proportion falls between 0.2 and 0.25?

Tip: When dealing with sample proportions, it's often useful to check if the conditions for normal approximation (np and n(1-p)) are met for a more accurate approximation.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Sample Proportion
Normal Distribution

Formulas

Standard Error for Proportion: SE = sqrt( (p(1-p)) / n )
Z-score formula for proportions: Z = (\hat{\pi} - p) / SE

Theorems

Central Limit Theorem
Normal Approximation to Binomial Distribution

Suitable Grade Level

Undergraduate Statistics or Advanced High School