Question

SKILL BUILDER 1 In Exercises 3.41 to \(3.44,\) data from a sample is being used to estimate something about a population. In each case: (a) Give notation for the quantity that is being estimated. (b) Give notation for the quantity that gives the best estimate. Random samples of people in Canada and people in Sweden are used to estimate the difference between the two countries in the proportion of people who have seen a hockey game (at any level) in the past year.

Short answer

The population parameter that is being estimated is \( P_C - P_S \), and the statistic that gives the best estimate is \( p_C - p_S \).

Step-by-step solution

Identification of the Population Parameter

The quantity that is being estimated from the problem mentioned is 'the difference between the proportions of people who have seen a hockey game in the past year in Canada and Sweden'. Let's represent this population parameter as \( P_C - P_S \), where \( P_C \) is the proportion of people who have seen a hockey game in Canada and \( P_S \) is the same in Sweden.

Identification of the Sample Statistic

We are using random samples from each country to estimate the population parameter. So the quantity that gives the best estimate would be the difference between the sample proportions from each country. Let's denote this sample statistic as \( p_C - p_S \), where \( p_C \) is the sample proportion of people who have seen a hockey game in Canada, and \( p_S \) is the same in Sweden.

Key concepts

Sample Proportion

When addressing the concept of sample proportion, we refer to the ratio of individuals in a sample meeting a certain criteria to the total number of individuals in that sample. For example, if a survey were conducted to determine how many people out of a selected group had seen a hockey game within the past year, the sample proportion (\( p \)) would be the number of people who had seen a game divided by the total number of surveyed individuals.

In the given exercise, sample proportions were denoted as \( p_C \) and \( p_S \) for Canada and Sweden respectively. It's important to gather a large enough sample size to ensure accuracy and diminish the impact of outliers or sampling errors, thus providing a reliable estimate of the population proportion.

Another point to note is that the accuracy of a sample as an estimate of the wider population depends significantly on how well the sample represents the entire population. This concept, known as representativeness, can greatly affect the reliability of the sample proportion as a proxy for the population proportion.

Population Proportion

While sample proportion refers to a subset, the population proportion encompasses the entire group of interest. In our hockey game example, the population proportion (\( P \) when referring to a general population) would be the actual ratio of all people in Canada or Sweden who have seen a hockey game in the last year.

Often, obtaining data for an entire population is impractical due to resource and time constraints, hence why we use sample data to estimate the population proportion. The exercise uses the notation \( P_C \) and \( P_S \) for the population proportions in Canada and Sweden, respectively.

Understanding the distinction between the sample and population proportions is fundamental in statistics as it guides how we interpret data and make predictions about larger groups based on sample findings. Sampling methods and techniques are tailored to maximize the likelihood that the sample proportion will accurately reflect the population proportion.

Difference Between Proportions

When we talk about the difference between proportions, we are comparing two sample or population proportions with the aim of understanding the relationship or disparity between them. The exercise presents a scenario requiring the estimation of the difference between the population proportions of people in Canada and Sweden who have seen a hockey game in the last year.

The formula used here for the difference between two population proportions is \( P_C - P_S \) while the difference between two sample proportions is represented as \( p_C - p_S \). This distinction is crucial since it helps us quantify the specific variation between groups from different populations.

In practical terms, calculating the difference between proportions can inform decisions, strategies, or policies relevant to the populations in question. For instance, understanding the difference in the popularity of hockey between Canada and Sweden could influence sports marketing strategies in both countries.

Confidence Interval

A confidence interval is a range of values that is likely to contain a population parameter with a certain level of confidence. Although not directly mentioned in the exercise, the concept is commonly used in estimation to account for the inherent uncertainty present when using sample data.

With reference to the previous concepts, a confidence interval would provide a range in which we believe the true difference between the population proportions (\( P_C - P_S \)) lies. It's calculated from our sample statistic (\( p_C - p_S \)) and accounts for variability within the data. The confidence level, usually expressed as a percentage (e.g., 95%), indicates the probability that the interval will capture the population parameter if we were to take many samples.

Understanding confidence intervals is essential when presenting and interpreting statistical estimates, as it provides context regarding the reliability of these estimates. It can be noted that wider confidence intervals suggest higher uncertainty in the estimate, while narrower intervals indicate more precise estimates, all else being equal.