Driving AgeAccording to a Gallup poll, 60% of U.S. women 18 years old or older stated that the minimum driving age should be 18. In a random sample of 15 U.S. women 18 years old or older, find the probability that: 


f. If a random sample of 200 U.S. women 18 years old or older resulted in 110 who believe that the minimum driving age should be 18, would this be unusual? Why?

Verified step by step guidance

1. Identify the parameter and sample information: The population proportion is \(p = 0.60\), the sample size is \(n = 200\), and the observed number of women who believe the minimum driving age should be 18 is \(x = 110\).
2. Calculate the sample proportion \(\hat{p}\) using the formula \(\hat{p} = \frac{x}{n}\). This gives the proportion of women in the sample who believe the minimum driving age should be 18.
3. Check the conditions for using the normal approximation to the binomial distribution: verify that both \(np\) and \(n(1-p)\) are at least 10 to ensure the sampling distribution of \(\hat{p}\) is approximately normal.
4. Calculate the mean and standard deviation of the sampling distribution of \(\hat{p}\): the mean is \(\mu_{\hat{p}} = p\), and the standard deviation is \(\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\).
5. Compute the z-score for the observed sample proportion using \(z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}\). Then, use the z-score to find the probability of observing a sample proportion as extreme as or more extreme than \(\hat{p}\). If this probability is very low (commonly less than 0.05), conclude that the result is unusual.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Here, 'success' is a woman believing the minimum driving age should be 18, with probability 0.60. It helps calculate probabilities for counts of such beliefs in samples.

Normal Approximation to the Binomial

For large sample sizes, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1-p). This simplifies probability calculations, especially for samples like 200 women, allowing use of z-scores to assess how unusual an observed count is.

Unusual Events and Standard Scores (Z-scores)

An event is considered unusual if its probability is very low, often corresponding to a z-score beyond ±2 standard deviations from the mean. Calculating the z-score for 110 women out of 200 helps determine if this result significantly deviates from the expected 60%, indicating whether the outcome is unusual.