MDM4U Introduction to Confidence IntervalsMs. Kueh

When you take a sample and calculate the mean of your sample _____, you are finding an estimate of the population mean _____. If you follow proper sampling techniques (from Chapter 2), your estimate is more likely to be representative of the population. However, there is always the possibility that your sample will include extreme members of the population, and thus your sample will not be representative of the population.

This is why most studies give their results in terms of a confidence interval. A confidence interval demonstrates:

The range of values that the population mean is likely to have The probability that the mean will be within that range of values

For example, if it is found that there is a 95% chance that the population mean will be between 1.311kg and 1.535kg, the confidence interval may be written in one of the following ways:

1. μ=1.423 kg±0.112 kg, 19 times out of 20

2. 1.311kg<μ<1.535kg (95% confidence interval)

3. The mean of 1.423 kg is accurate to within ±8 %, 19 times out of 20

In the above examples, the values

1.423 kg represents the _______________________. This is the researcher’s estimate of the population mean.

0.112kg and ±8 % represent the ____________________________ for the study. This depends on a number of factors, including the population size, the standard deviation of the variable, and the sample size.

“19 times out of 20” or 95 % is the _________________________________. This shows how likely it is that the actual population mean is within the range given by the confidence interval. A 95 % confidence interval means that there is a 5 % chance that your confidence interval will not include the actual population mean.

Notes:

1. Even though we’ve given a range for the population mean, we can never know for sure what the population mean is, so we can’t even say that it’s necessarily within a particular range. We can only say that there is a 95% or 99% or 90% probability that it is within a certain range.

2. The most commonly used confidence level is 95%. If you want to be particularly accurate in your study, such as in a trial of a new drug, you might use a confidence level of 99%. If you are less concerned about accuracy, such as for an opinion poll, you might use a confidence level of 90%.

Homework1. For each of the following, determine:

i) Sample mean

ii) Margin of error (as a % and a measurement)

iii) Range of values within the confidence interval

iv) Probability that the actual mean being within the confidence interval

a) = 89.22 L 8.24 L, 9 times out of 10

b) 5.332 m < < 8.456 m (99% confidence interval)

c) = 43.22 kg 5%, 19 times out of 20.

d) 1.821 m < < 5.122 m (90% confidence interval)

2. Interpret each of the following statements using confidence intervals.a. In a recent survey, 42% of high school graduates indicated that they expected to

earn over $100, 000 per year by the time they retire. This survey is considered accurate within plus or minus 3%, 19 times in 20.

b. A survey done by the incumbent MP indicated that 48% of decided voters said they would vote for him again in the next election. The result is considered accurate within plus or minus 5%, nine times in ten.

c. According to a market research firm, 28% of teenagers will purchase the latest CD by rock band Drench. The result is considered accurate within 4%, 11 out of 15 times.

3. Ms. Robinson used simple random sampling of ten students in her Calculus class. She found that the average mark was 80%. The actual average mark in the class is 77%.

a. Assuming she used proper sampling technique, how is this possible?

b. How could she get a more accurate result from her survey without altering her sampling method?

What level of confidence (and why) would you use for a study looking at:

a. What people are wearing to the prom

b. Who people are going to vote for in the next federal election

c. Whether or not a new heart medicine causes cancer

d. Whether carcinogens are being released by a waste facility

Solutions:1a) i. 89.22 L ii. 8.24L, 9.2%

iii. 80.98 L < < 97.46 L iv. 90% C.I.

1 b) i. 6.894 m ii. 1.562 m, 22.7%iii. 5.332 m < < 8.456 m iv. 99% C.I.

1 c) i. 43.22 kg ii. 2.161 kg, 5%iii. 41.059 kg < < 45.381 kg iv. 95% C.I.

1 d) i. 3.472 m ii. 1.651 m, 47.5%iii. 1.821 m < < 5.122 m iv. 90% C.I.

2. a) There is a 95% probability that between 39% and 45% of high school graduates expect to earn over $100 000 per year by the time they retire

b) There is a 90% probability that between 43% and 53% of decided voters will vote for the incumbent in the next election.

c) There is a 73% probability that between 24% and 32% of teenagers will purchase the latest CD by the rock band.

3a) Since she is taking a sample of the population, she might sample some from a higher range (further from the mean). When sampling from a single sample, statistics such as the mean may differ from those of the underlying population.

3b) When repeated samples of the same size are drawn from a normal population, the sample means will be normally distributed with a mean equal to the population mean. Alternatively, she could take a larger sample size.

4. a) 90% C.I. b) 95% C.I. c) 99% C.I. d) 99% C.I.