Roper ASW conducted a survey to learn about American adults'...

Roper ASW conducted a survey to learn about American adults' attitudes toward money and happiness (Money, October $2003 ) .$ Fifty-six percent of the respondents said they balance their checkbook at least once a month. a. Suppose a sample of 400 American adults were taken. Show the sampling distribution of the proportion of adults who balance their checkbook at least once a month. b. What is the probability the the sample proportion will be within $\pm .02$ of the population proportion? c. What is the probability that the sample proportion will be within $\pm .04$ of the population proportion?

Roper ASW conducted a survey to learn about American adults' attitudes toward money and happiness (Money, October $2003 ) .$ Fifty-six percent of the respondents said they balance their checkbook at least once a month.
a. Suppose a sample of 400 American adults were taken. Show the sampling distribution of the proportion of adults who balance their checkbook at least once a month.
b. What is the probability the the sample proportion will be within $\pm .02$ of the population proportion?
c. What is the probability that the sample proportion will be within $\pm .04$ of the population proportion?

Key Concepts

Z-scores and Probability Calculation

Z-scores represent the number of standard errors a particular sample statistic is away from the population parameter. They are used in conjunction with the normal approximation to determine the probability that the sample proportion falls within a desired range of the true proportion.

Sampling Distribution of a Proportion

This concept refers to the distribution of values taken by the proportion of successes in a sample, calculated from repeated random samples of a fixed size from a population. It is centered at the true population proportion and its variability decreases as the sample size increases.

Central Limit Theorem

The Central Limit Theorem is fundamental in inferential statistics; it states that, given a sufficiently large sample size, the sampling distribution of a proportion will be approximately normally distributed regardless of the population’s distribution. This enables the use of normal-based methods in probability calculations.

Standard Error of a Sample Proportion

The standard error measures the spread of the sampling distribution and is given by the formula sqrt[p(1-p)/n], where p is the population proportion and n is the sample size. It quantifies the expected fluctuation of the sample proportion around the population proportion.

Normal Approximation

For large samples, the sampling distribution of a sample proportion can be approximated by a normal distribution. This approximation facilitates the calculation of probabilities and confidence intervals for the proportion by using properties of the normal distribution.

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