"Spanking
In March 1995, The Harris Poll reported that 80% of parents spank their children. Suppose a recent poll of 1030 adult Americans with children finds that 781 indicated that they spank their children. If we assume parents’ attitude toward spanking has not changed since 1995, how many of 1030 parents surveyed would we expect to spank?
Do the results of the survey suggest that parents’ attitude toward spanking may have changed since 1995? Why?"

Verified step by step guidance

1. Identify the population proportion from the 1995 Harris Poll, which is given as \(p = 0.80\) (80% of parents spank their children).
2. Calculate the expected number of parents who would spank their children in the recent survey of 1030 parents, assuming the proportion has not changed. Use the formula: \(\text{Expected count} = p \times n\), where \(n = 1030\).
3. Compare the observed number of parents who spank their children in the recent survey, which is 781, to the expected count calculated in the previous step.
4. To determine if the parents' attitude toward spanking has changed, consider performing a hypothesis test for a population proportion. Set up the null hypothesis \(H_0: p = 0.80\) and the alternative hypothesis \(H_a: p \neq 0.80\).
5. Calculate the test statistic (z-score) using the formula: \(z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}\), where \(\hat{p} = \frac{781}{1030}\) is the sample proportion. Then, compare the z-score to critical values or use a p-value to decide whether to reject the null hypothesis, which will indicate if there is evidence that the attitude has changed.

Key Concepts

Expected Value in Probability

The expected value represents the average outcome we anticipate based on a given probability. In this context, if 80% of parents spank their children, the expected number out of 1030 parents is 0.80 × 1030 = 824. This helps us compare observed data to what we would expect if attitudes remained unchanged.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine if there is enough evidence to reject a presumed claim. Here, the null hypothesis assumes the spanking rate is still 80%. We test whether the observed number (781) significantly differs from the expected number (824) to infer if attitudes have changed.

Sampling Variability and Significance

Sampling variability refers to natural fluctuations in sample results due to chance. Even if the true rate is 80%, the observed number may differ. Statistical significance tests help decide if the difference between observed and expected values is due to chance or indicates a real change in attitudes.