A local high school records data about each graduating class. The following table shows graduation and enrollment numbers for members of three given graduation years. 1 1.5.1: Estimating Sample Proportions Class Graduate d Still enrolled No longer enrolled 2012 1,042 345 221 2011 1,120 306 198 2010 1,025 450 222 1. What is the graduation rate for each class: 2012, 2011, and 2010? Round your answer to the nearest percent. 2. What is the high school’s graduation rate over the three given years? 3. Which class best represents the overall graduation rate? Explain your reasoning.
Transcript
A local high school records data about each graduating class. The following table shows graduation and enrollment numbers for members of three given graduation years.
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1.5.1: Estimating Sample Proportions
Class Graduated Still enrolled No longer enrolled2012 1,042 345 2212011 1,120 306 1982010 1,025 450 222
1. What is the graduation rate for each class: 2012, 2011, and 2010? Round your answer to the nearest percent.
2. What is the high school’s graduation rate over the three given years?
3. Which class best represents the overall graduation rate? Explain your reasoning.
1. What is the graduation rate for each class: 2012, 2011, and 2010? Round your answer to the nearest percent.
• To find the graduation rate for each class, first calculate the total number of students in the class.
• To find the graduation rate for a given class, divide the number of students who graduated by the total number of students in the class.
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1.5.1: Estimating Sample Proportions
Key Concepts• Sometimes data sets are too large to measure. When
we cannot measure the entire data set, called a population, we take a sample or a portion of the population to measure.
• A sample population is a portion of the population. The number of elements or observations in the sample population is denoted by n.
• The sample proportion is the name we give for the estimate of the population, based on the sample data that we have. This is often represented by , which is pronounced “p hat.”
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1.5.1: Estimating Sample Proportions
Key Concepts, continued
• The sample proportion is calculated using the formula
, where p is the number of favorable outcomes
and n is the sample population.
• When expressing a sample proportion, we can use a fraction, a percentage, or a decimal.
• Favorable outcomes, also known as desirable outcomes or successes, are those data sought or hoped for in a survey, but are not limited to these data; favorable outcomes also include the percentage of people who respond to a survey.
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1.5.1: Estimating Sample Proportions
Key Concepts, continued
• The standard error of the proportion (SEP) is the
variability of the measure of the proportion of a sample.
The formula used to calculate the standard error of the
proportion is , where is the sample
proportion determined by the sample and n is the number
of elements or observations in the sample population. 5
1.5.1: Estimating Sample Proportions
Key Concepts, continued• This formula is valid when the population is at least 10
times as large as the sample. Such a size ensures that the population is large enough to estimate valid conclusions based on a random sample.
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1.5.1: Estimating Sample Proportions
Guided Practice
Example 3If 540 out of 3,600 high school graduates who answer a post-graduation survey indicate that they intend to enter the military, what is the standard error of the proportion for this sample population to the nearest hundredth?
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1.5.1: Estimating Sample Proportions
Guided Practice: Example 3, continued
1. Identify the given information.
The number of favorable outcomes, p, is 540.
The number of elements in the sample population, n, is 3,600.
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1.5.1: Estimating Sample Proportions
Guided Practice: Example 3, continued
2. Calculate the sample proportion.
Use the formula for calculating the sample proportion:
, where p is the number of favorable outcomes
and n is the number of elements in the sample
population.
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1.5.1: Estimating Sample Proportions
Guided Practice: Example 3, continued
Substitute the known quantities.
Sample proportion formula
Substitute 540 for p and 3,600 for n.
Simplify.
The sample proportion is 0.15.
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1.5.1: Estimating Sample Proportions
Guided Practice: Example 3, continued
3. Calculate the standard error of the proportion.
Use the formula for calculating the standard error of
the proportion: , where n is the
number of elements in the sample population and is
the sample proportion.
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1.5.1: Estimating Sample Proportions
Guided Practice: Example 3, continued
Formula for the standard error of the proportion
Substitute 0.15 for and 3,600 for n.
Simplify.
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1.5.1: Estimating Sample Proportions
Guided Practice: Example 3, continued
Continue to simplify.
Substitute 0.15 for and 3,600 for n.
SEP ≈ 0.00595119
SEP ≈ 0.01 Round to the nearest hundredth.
The standard error of the proportion is approximately 0.01. 13
1.5.1: Estimating Sample Proportions
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Example 1
A sample of 480 townspeople were surveyed about their opinions of an elected official’s decisions. If 336 responded in support of the official’s decisions, what is the sample proportion, ˆp , for the official’s approval rating amongst this sample population?
Estimate the standard error of the population
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Guided Practice
Example 4Shae owns a carnival and is testing a new game. She would like the game to have a 50% win rate, with 0.05 for the standard error of the proportion. How many times should Shae test the game to ensure these numbers?
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1.5.1: Estimating Sample Proportions
Guided Practice: Example 4, continued
1. Identify the given information. Shae would like the game to have a 50% win rate; therefore, the sample proportion, , is 50% or 0.5.
The standard error of the proportion is given as 0.05.
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1.5.1: Estimating Sample Proportions
Guided Practice: Example 4, continued
2. Determine the sample population.
Use the formula for calculating the standard error of
the proportion: , where n is the
number of elements in the sample population and is
the sample proportion.
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1.5.1: Estimating Sample Proportions
Guided Practice: Example 4, continued
Formula for the standard
error of the proportion
Substitute 0.05 for SEP and 0.5 for
.
Simplify.
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1.5.1: Estimating Sample Proportions
Guided Practice: Example 4, continuedSolve the equation for n, the number of elements in the sample population.
Square both sides of the
equation.
Simplify.
0.0025n = 0.25 Multiply both sides by n.
n = 100 Divide both sides by 0.0025.
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1.5.1: Estimating Sample Proportions
Guided Practice: Example 4, continued
The number of elements in the sample population, n, is 100; therefore, Shae should test the game 100 times to ensure a 50% win rate and a standard error of 0.05.
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1.5.1: Estimating Sample Proportions
✔
The police chief of a small town wants to add surveillance cameras at all the traffic lights in the town to cut down on accidents. He surveyed some community members, and found that 16 out of 24 people favored the cameras. When the chief shared this data at a town council meeting, a councilor who works as a statistician objected to the small sample size. She said she would not vote in favor of surveillance cameras until the standard error of the proportion for the sample population is reduced to less than 0.03.
The police chief plans to conduct a new survey to fulfill the councilor’s request. If the sample proportion of the new survey remains consistent with that of the first survey, how many people must be sampled in order for the councilor’s request to be granted? Pg 139 in your workbook – answer questions on page 140. 21
