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“Metadata Description of Leadership PAC List.” Federal Election Commission. Available online at http://www.fec.gov/finance/disclosure/metadata/metadataLeadershipPacList.shtml (accessed July 2, 2013).

Chapter review

In many cases, the researcher does not know the population standard deviation, σ , of the measure being studied. In these cases, it is common to use the sample standard deviation, s , as an estimate of σ . The normal distribution creates accurate confidence intervals when σ is known, but it is not as accurate when s is used as an estimate. In this case, the Student’s t-distribution is much better. Define a t-score using the following formula:

t =   x ¯   μ s n

The t -score follows the Student’s t-distribution with n – 1 degrees of freedom. The confidence interval under this distribution is calculated with EBM = ( t α 2 ) s n where t α 2 is the t -score with area to the right equal to α 2 , s is the sample standard deviation, and n is the sample size. Use a table, calculator, or computer to find t α 2 for a given α .

Formula review

s = the standard deviation of sample values.

t =   x ¯ μ s n is the formula for the t -score which measures how far away a measure is from the population mean in the Student’s t-distribution

df = n - 1; the degrees of freedom for a Student’s t-distribution where n represents the size of the sample

T ~ t df the random variable, T , has a Student’s t-distribution with df degrees of freedom

E B M = t α 2 s n = the error bound for the population mean when the population standard deviation is unknown

t α 2 is the t -score in the Student’s t-distribution with area to the right equal to α 2

The general form for a confidence interval for a single mean, population standard deviation unknown, Student's t is given by (lower bound, upper bound)
= (point estimate – EBM , point estimate + EBM )
= ( x ¯ t s n , x ¯ t s n )

Use the following information to answer the next five exercises. A hospital is trying to cut down on emergency room wait times. It is interested in the amount of time patients must wait before being called back to be examined. An investigation committee randomly surveyed 70 patients. The sample mean was 1.5 hours with a sample standard deviation of 0.5 hours.

Identify the following:

  1. x ¯ =_______
  2. s x =_______
  3. n =_______
  4. n – 1 =_______
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Define the random variables X and X ¯ in words.

X is the number of hours a patient waits in the emergency room before being called back to be examined. X ¯ is the mean wait time of 70 patients in the emergency room.

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Which distribution should you use for this problem?

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Construct a 95% confidence interval for the population mean time spent waiting. State the confidence interval, sketch the graph, and calculate the error bound.

CI: (1.3808, 1.6192)

This is a normal distribution curve. The peak of the curve coincides with the point 1.5 on the horizontal axis.  A central region is shaded between points 1.38 and 1.62.

EBM = 0.12

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Explain in complete sentences what the confidence interval means.

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Use the following information to answer the next six exercises: One hundred eight Americans were surveyed to determine the number of hours they spend watching television each month. It was revealed that they watched an average of 151 hours each month with a standard deviation of 32 hours. Assume that the underlying population distribution is normal.

Identify the following:

  1. x ¯ =_______
  2. s x =_______
  3. n =_______
  4. n – 1 =_______
  1. x ¯ = 151
  2. s x = 32
  3. n = 108
  4. n – 1 = 107
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Define the random variable X in words.

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Define the random variable X ¯ in words.

X ¯ is the mean number of hours spent watching television per month from a sample of 108 Americans.

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Which distribution should you use for this problem?

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Construct a 99% confidence interval for the population mean hours spent watching television per month. (a) State the confidence interval, (b) sketch the graph, and (c) calculate the error bound.

CI: (142.92, 159.08)

This is a normal distribution curve. The peak of the curve coincides with the point 151 on the horizontal axis.  A central region is shaded between points 142.92 and 159.08.

EBM = 8.08

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Why would the error bound change if the confidence level were lowered to 95%?

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Use the following information to answer the next 13 exercises: The data in [link] are the result of a random survey of 39 national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true mean number of colors on a national flag. Let X = the number of colors on a national flag.

X Freq.
1 1
2 7
3 18
4 7
5 6

Calculate the following:

  1. x ¯ =______
  2. s x =______
  3. n =______
  1. 3.26
  2. 1.02
  3. 39
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Define the random variable X ¯ in words.

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What is x ¯ estimating?

μ

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As a result of your answer to [link] , state the exact distribution to use when calculating the confidence interval.

t 38

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Construct a 95% confidence interval for the true mean number of colors on national flags.

How much area is in both tails (combined)?

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How much area is in each tail?

0.025

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Calculate the following:

  1. lower limit
  2. upper limit
  3. error bound

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The 95% confidence interval is_____.

(2.93, 3.59)

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Fill in the blanks on the graph with the areas, the upper and lower limits of the Confidence Interval and the sample mean.

This is a template of a normal distribution curve with the central region shaded to represent a confidence interval. The residual areas are on either side of the shaded region. Blanks indicate that students should label the confidence level, residual areas, and points that define the confidence interval.
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In one complete sentence, explain what the interval means.

We are 95% confident that the true mean number of colors for national flags is between 2.93 colors and 3.59 colors.

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Using the same x ¯ , s x , and level of confidence, suppose that n were 69 instead of 39. Would the error bound become larger or smaller? How do you know?

The error bound would become EBM = 0.245. This error bound decreases because as sample sizes increase, variability decreases and we need less interval length to capture the true mean.

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Using the same x ¯ , s x , and n = 39, how would the error bound change if the confidence level were reduced to 90%? Why?

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Questions & Answers

mean is number that occurs frequently in a giving data
Chinedu Reply
That places the mode and the mean as the same thing. I'd define the mean as the ratio of the total sum of variables to the variable count, and it assigns the variables a similar value across the board.
Samsicker
what is mean
John Reply
what is normal distribution
RAHAT Reply
What is the uses of sample in real life
Waqas Reply
change of origin and scale
RAHAT Reply
3. If the grades of 40000 students in a course at the Hashemite University are distributed according to N(60,400) Then the number of students with grades less than 75 =*
Ahmad Reply
If a constant value is added to every observation of data, then arithmetic mean is obtained by
Madiha Reply
sum of AM+Constnt
Fazal
data can be defined as numbers in context. suppose you are given the following set of numbers 18,22,22,20,19,21
Tyasia Reply
what are data
Tyasia Reply
what is mode?
Natasha Reply
what is statistics
Natasha
statistics is a combination of collect data summraize data analyiz data and interprete data
Ali
what is mode
Natasha
what is statistics
Alex Reply
It is the science of analysing numerical data in large quantities, especially for the purpose of inferring proportions in a whole from those in a representative sample.
Bernice
history of statistics
Terseer Reply
statistics was first used by?
Terseer
if a population has a prevalence of Hypertension 5%, what is the probability of 4 people having hypertension from 8 randomly selected individuals?
John Reply
Carpet land sales persons average 8000 per weekend sales Steve qantas the firm's vice president proposes a compensation plan with new selling incentives Steve hopes that the results of a trial selling period will enable him to conclude that the compensation plan increases the average sales per sales
lorenda Reply
Supposed we have Standard deviation 1.56, mean 6.36, sample size 25 and Z-score 1.96 at 95% confidence level, what is the confidence interval?
John Reply

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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