# 8.2 A single population mean using the student t distribution  (Page 4/21)

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## 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:

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 = $\left({t}_{\frac{\alpha }{2}}\right)\frac{s}{\sqrt{n}}$ where ${t}_{\frac{\alpha }{2}}$ is the t -score with area to the right equal to $\frac{\alpha }{2}$ , s is the sample standard deviation, and n is the sample size. Use a table, calculator, or computer to find ${t}_{\frac{\alpha }{2}}$ for a given α .

## Formula review

s = the standard deviation of sample values.

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

$EBM={t}_{\frac{\alpha }{2}}\frac{s}{\sqrt{n}}$ = the error bound for the population mean when the population standard deviation is unknown

${t}_{\frac{\alpha }{2}}$ is the t -score in the Student’s t-distribution with area to the right equal to $\frac{\alpha }{2}$

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 )
=

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. $\overline{x}$ =_______
2. ${s}_{x}$ =_______
3. n =_______
4. n – 1 =_______

Define the random variables X and $\overline{X}$ in words.

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

Which distribution should you use for this problem?

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)

EBM = 0.12

Explain in complete sentences what the confidence interval means.

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. $\overline{x}$ =_______
2. ${s}_{x}$ =_______
3. n =_______
4. n – 1 =_______
1. $\overline{x}$ = 151
2. ${s}_{x}$ = 32
3. n = 108
4. n – 1 = 107

Define the random variable X in words.

Define the random variable $\overline{X}$ in words.

$\overline{X}$ is the mean number of hours spent watching television per month from a sample of 108 Americans.

Which distribution should you use for this problem?

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)

EBM = 8.08

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

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. $\overline{x}$ =______
2. ${s}_{x}$ =______
3. n =______
1. 3.26
2. 1.02
3. 39

Define the random variable $\overline{X}$ in words.

What is $\overline{x}$ estimating?

μ

Is ${\sigma }_{x}$ known?

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

t 38

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

How much area is in both tails (combined)?

How much area is in each tail?

0.025

Calculate the following:

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

The 95% confidence interval is_____.

(2.93, 3.59)

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

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.

Using the same $\overline{x}$ , ${s}_{x}$ , 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.

Using the same $\overline{x}$ , ${s}_{x}$ , and n = 39, how would the error bound change if the confidence level were reduced to 90%? Why?

mean is number that occurs frequently in a giving data
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
what is normal distribution
What is the uses of sample in real life
change of origin and scale
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 =*
If a constant value is added to every observation of data, then arithmetic mean is obtained by
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
what are data
what is mode?
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
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
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?
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
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?