A recent survey in the N . Y. Times Almanac indicated that...

A recent survey in the $N .$ Y. Times Almanac indicated that $48.8 \%$ of families own stock. A broker wanted to determine if this survey could be valid. He surveyed a random sample of 250 families and found that 142 owned some type of stock. At the 0.05 significance level, can the survey be considered to be accurate?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical procedure that involves making an assumption (the null hypothesis) about a population parameter and then using sample data to determine whether there is enough evidence to reject that assumption in favor of an alternative hypothesis. This method helps in deciding if observed effects are statistically significant or possibly due to random chance.

Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion refers to the probability distribution of the proportion statistic obtained from repeated random samples of a fixed size from the population. Under the Central Limit Theorem, if the sample size is sufficiently large, this distribution approximates a normal distribution, which allows for the use of normal-based inference techniques for estimating population parameters.

Test Statistic for a Proportion

The test statistic for a proportion is calculated by comparing the difference between the observed sample proportion and the hypothesized population proportion relative to the standard error of the sample proportion. This typically results in a z-score, which quantifies how far, in standard deviation units, the sample proportion is from the hypothesized value under the null distribution.

Significance Level (Type I Error)

The significance level, denoted by alpha (?), is the threshold set by the researcher for determining whether the observed data is sufficiently unlikely under the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true (a Type I error), and common values such as 0.05 indicate a 5% risk of this error occurring.

p-value

The p-value is a measure that indicates the probability of observing a test statistic as extreme as, or more extreme than, the one obtained from the sample data assuming the null hypothesis is correct. If the p-value is less than the predetermined significance level, it suggests that the sample provides sufficient evidence to reject the null hypothesis, supporting the alternative hypothesis.

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