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Mathematics 20-2 Module 4

Module 4: Statistical Reasoning

Explore

In statistics, level of confidence is used to show how certain statiticians are that the data captures the true meaning of whatever is being tested. Level of confidence also gives an indication of whether or not the results are repeatable. In other words, if the survey or test was repeated, how certain could you be that you would get the same results over and over again?

This is a health warning label found on a cigarette package. It shows a type of mouth cancer caused by smoking.

Health Canada

Consider the following survey results.

In January 2011, a Vision Critical/Angus Reid poll of 1022 Canadian adults found that 82% of repondents support the use of health warnings on cigarette packages. The results of the poll were accuate within plus or minus 3.1% 19 times out of 20.

margin of error: the measurement of the accuracy of the results of a survey

The larger the margin of error, the less accurate the estimated value. Margin of error is usually expressed as plus or minus a percent. Example: ±5%

confidence interval: interval in which the true value you are trying to determine would be expected to lie, to a stated degree of accuracy

Confidence interval can be expressed using plus or minus notation or as a range of values. Example: 55% ± 5% or 50% to 60%

confidence level: likelihood that the result for the “true” population lies within the confidence interval

A confidence level of 95% is usually used for surveys, but 90% or 99% is sometimes used.

The poll tells us that the companies doing the study know that the survey results are not 100% accurate. The companies are confident that the actual percentage of people who share this belief is unlikely to be 3.1% higher than 82% or 3.1% lower than 82%; this is known as a margin of error. The pollsters can accurately say that if you repeated this survey on the entire adult population, you would get results that were within 3.1% of their results 19 times out of 20.

So, the pollsters can say with some confidence that between 78.9% (82% − 3.1%) and 85.1% (82% + 3.1%) of the population would support the use of health warnings on cigarette packages. This range between 78.9% and 85.1% is called the confidence interval. The confidence interval is determined by taking the survey or poll result (82%) plus or minus the margin of error (3.1%).

This illustration shows how the margin of error can be used to determine the confidence interval of a survey.

The confidence level is 19 out of 20 or 95%. For this survey, you can be 95% certain that between 78.9% and 85.1% of the “true” population would be in support of the health labels.

The confidence interval can be used to estimate the range of the mean for the population. In other words, it can help you estimate how many people in the population would support health labels on cigarettes. Suppose the total population of adults is 50 000. According to the survey results, between 78.9% and 85.1% of the population would support health labels.

50 000 × 78.9% = 39 450

50 000 × 85.1% = 42 550

So, you can say with 95% confidence that between 39 450 to 42 550 adults, in a population of 50 000 adults, would support the use of health labels on cigarettes.

Read “Example 1: Analyzing and applying survey results” on page 295 of your textbook to see another example of how confidence intervals can be used to determine the certainty of a survey’s results.

Try This 2

You will submit your work from Try This 2 to your teacher for marking.

The results of the Vision Critical/Angus Reid poll also found that 60% of Canadians believe the images currently used for health warnings on cigarette packages are about right, while 24% wanted even more graphic imagery.

Calculate the range of people that would want more graphic images in a population of 30 000 adults. Recall that the margin of error for the survey was ±3.1% and the confidence level was 95% (19/20). (3 marks)