Question

It probably wouldn't surprise you to know that Valentine's Day means big business for florists, jewelry stores, and restaurants. But did you know that it is also a big day for pet stores? In January \(2010,\) the National Retail Federation conducted a survey of consumers in a representative sample of adult Americans ("This Valentine's Day, Couples Cut Back on Gifts to Each Other, According to NRF Survey," www.nrf.com). One of the questions in the survey asked if the respondent planned to spend money on a Valentine's Day gift for his or her pet. a. The proportion who responded that they did plan to purchase a gift for their pet was 0.173 . Suppose that the sample size for this survey was \(n=200\). Construct and interpret a \(95 \%\) confidence interval for the proportion of all adult Americans who planned to purchase a Valentine's Day gift for their pet. b. The actual sample size for the survey was much larger than 200\. Would a \(95 \%\) confidence interval calculated using the actual sample size have been narrower or wider than the confidence interval calculated in Part (a)?

Short answer

a) The 95% confidence interval for the sample proportion is obtained using the given sample size and the sample proportion. It provides the range in which the true population proportion is likely to fall. b) If the sample size were larger, the 95% confidence interval would be narrower, as increased sample size decreases the standard error and thus the margin of error.

Step-by-step solution

Calculation of interval

We first find the confidence interval for the given sample size of 200. The proportion is given as 0.173, and we use a z-value of about 1.96 for a 95% confidence interval. Substituting these values in, we obtain the interval as \(0.173 \pm 1.96 \sqrt{ \frac{(0.173)(0.827)}{200}}\)

Calculation of lower and upper interval

Now calculate the lower and upper ends of the interval. The margin of error is equal to \(1.96 \sqrt{ \frac{(0.173)(0.827)}{200}}\). Subtract this from 0.173 to get the lower end, and add this to 0.173 to get the upper end. These will be the lower and upper ends of the confidence interval respectively.

Interpretation of Confidence Interval

The calculated interval is the range in which we are 95% confident that the true population proportion lies. This means that we can be quite certain that the actual proportion of all adult Americans planning to buy their pet a Valentine's Day gift is within this range.

Effect of Larger Sample Size

If the sample size were larger than 200, the confidence interval would be narrower than the one computed in part (a). This is because the standard error, which is in the denominator of the formula for margin of error, would be smaller as the sample size increases. Hence, the margin of error would decrease, making the confidence interval more precise.

Key concepts

Survey Sample Size

Understanding the role of survey sample size is crucial when conducting research. Sample size refers to the number of individuals or observations included in a survey. It significantly impacts the accuracy and reliability of the survey results.

A sizable sample size generally leads to more reliable results because it is more representative of the population. However, as the size increases, the cost and time spent on data collection also increase. In statistical terms, sample size is directly related to the confidence interval and the standard error. A larger sample size will reduce the standard error and yield a narrower confidence interval, indicating a more precise estimation of the population parameter.

In the context of our original exercise, the National Retail Federation conducted a survey to determine the proportion of adult Americans planning to buy their pets a Valentine's Day gift. For a sample size of 200, we calculated a 95% confidence interval for the estimated proportion. If their actual survey included significantly more than 200 respondents, the results could be more precise, reflecting a narrower confidence interval.

Population Proportion Estimation

Estimating the population proportion is a common objective in surveys. When gathering data from a sample, the goal is to make inferences about the larger population. The population proportion represents the fraction of people in the entire population who exhibit a particular quality or characteristic.

In the educational exercise, we estimated the proportion of all adult Americans who plan to purchase a Valentine's Day gift for their pets based on the survey results. This estimation is valuable as it gives businesses and economists insights into consumer behavior and spending habits. The estimated proportion of 0.173 from the survey is a point estimate and serves as the best single guess of the true population proportion. To assess the estimate’s reliability, we calculate a confidence interval around it.

The process involves determining a range of values within which the true population proportion is likely to fall. The level of confidence (usually expressed as 90%, 95%, or 99%) describes the expected probability that this interval captures the actual population proportion.

Standard Error

The concept of standard error is pivotal in understanding the precision of a sample estimate. It measures the variability of a statistic from sample to sample if you were to repeat the survey many times. Essentially, it's an indication of how much a sample's statistic (like the mean or proportion) is likely to fluctuate due to random sampling error.

The standard error is calculated from the standard deviation and the sample size. It is inversely related to the square root of the sample size; thus, as the sample size increases, the standard error decreases. A smaller standard error means that the sample statistic is more closely clustered around the population parameter it is estimating, suggesting more precision.

In the original exercise, the standard error is used to construct the 95% confidence interval for the survey estimate. We can see that the confidence interval depends on both the sample proportion and the standard error, and as the sample size grows, the standard error diminishes, leading to a tighter confidence interval, which increases our trust in the precision of the population proportion estimate.