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Student learning outcomes

By the end of this chapter, the student should be able to:

  • Calculate and interpret confidence intervals for one population mean and one population proportion.
  • Interpret the student-t probability distribution as the sample size changes.
  • Discriminate between problems applying the normal and the student-t distributions.

Introduction

Suppose you are trying to determine the mean rent of a two-bedroom apartment in your town. You might look in the classified section of the newspaper, writedown several rents listed, and average them together. You would have obtained a point estimate of the true mean. If you are trying to determine the percent of timesyou make a basket when shooting a basketball, you might count the number of shots you make and divide that by the number of shots you attempted. In thiscase, you would have obtained a point estimate for the true proportion.

We use sample data to make generalizations about an unknown population. This part of statistics is called inferential statistics . The sample data help us to make an estimate of a population parameter . We realize that the point estimate is most likely not the exact value of the population parameter, but close to it. Aftercalculating point estimates, we construct confidence intervals in which we believe the parameter lies.

In this chapter, you will learn to construct and interpret confidence intervals. You will also learn a new distribution, the Student's-t, and how it is used with theseintervals. Throughout the chapter, it is important to keep in mind that the confidence interval is a random variable. It is the parameter that isfixed.

If you worked in the marketing department of an entertainment company, you might be interested in the mean number of compact discs (CD's) a consumerbuys per month. If so, you could conduct a survey and calculate the sample mean, x , and the sample standard deviation, s . You would use x to estimate the population mean and s to estimate the population standard deviation. The sample mean, x , is the point estimate for the population mean, μ . The sample standard deviation, s , is the point estimate for the population standard deviation, σ .

Each of x and s is also called a statistic.

A confidence interval is another type of estimate but, instead of being just one number, it is an interval of numbers. The interval of numbers is a range of values calculated from a given set of sample data. The confidence interval is likely to include an unknown population parameter.

Suppose for the CD example we do not know the population mean μ but we do know that the population standard deviation is σ = 1 and our sample size is 100. Then by the Central Limit Theorem, the standard deviation for the sample mean is

σ n = 1 100 = 0.1 .

The Empirical Rule , which applies to bell-shaped distributions, says that in approximately 95% of the samples, the sample mean, x , will be within two standard deviations of the population mean μ . For our CD example, two standard deviations is (2)(0.1) = 0.2 . The sample mean x is likely to be within 0.2 units of μ .

Because x is within 0.2 units of μ , which is unknown, then μ is likely to be within 0.2 units of x in 95% of the samples. The population mean μ is contained in an interval whose lower number is calculated by taking the sample mean and subtractingtwo standard deviations ( ( 2 ) ( 0.1 ) ) and whose upper number is calculated by taking the sample mean and adding two standard deviations. In other words, μ is between x - 0.2 and x + 0.2 in 95% of all the samples.

For the CD example, suppose that a sample produced a sample mean x = 2 . Then the unknown population mean μ is between

x - 0.2 = 2 - 0.2 = 1.8 and x + 0.2 = 2 + 0.2 = 2.2

We say that we are 95% confident that the unknown population mean number of CDs is between 1.8 and 2.2. The 95% confidence interval is (1.8, 2.2).

The 95% confidence interval implies two possibilities. Either the interval (1.8, 2.2) contains the true mean μ or our sample produced an x that is not within 0.2 units of the true mean μ . The second possibility happens for only 5% of all the samples (100% - 95%).

Remember that a confidence interval is created for an unknown population parameter like the population mean, μ . Confidence intervals for some parameters have the form

(point estimate - margin of error, point estimate + margin of error)

The margin of error depends on the confidence level or percentage of confidence.

When you read newspapers and journals, some reports will use the phrase "margin of error." Other reports will not use that phrase, but include a confidence interval as the point estimate + or - the margin oferror. These are two ways of expressing the same concept.

Although the text only covers symmetric confidence intervals, there are non-symmetric confidence intervals (for example, a confidence interval for the standard deviation).

Optional collaborative classroom activity

Have your instructor record the number of meals each student in your class eats out in a week. Assume that the standard deviation is known to be 3 meals.Construct an approximate 95% confidence interval for the true mean number of meals students eat out each week.

  1. Calculate the sample mean.
  2. σ = 3 and n = the number of students surveyed.
  3. Construct the interval ( x - 2 σ n , x + 2 σ n )

We say we are approximately 95% confident that the true average number of meals that students eat out in a week is between __________ and ___________.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
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David
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emma Reply
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what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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answer
Magreth
progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
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Source:  OpenStax, Collaborative statistics (custom online version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11476/1.5
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