# 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?

what is statistics
statistics is the beach of mathematics which deals with collection ,organisation, presentation, analysis and interpretation of numerical data
Saeed
oh but interpretation of data, like what and how? 🤔
Bhavani
interpretation: Think in a way that you have given a company year turnover and you have a record of 100years and data set is like (Year,Turnover). Now with that data you can interpret many thing how was the company growth, when were the losses and other things
Akash
interpretation: it is a process in which we make a decision about a population on the basis of sample data . example: if we want to interpret the average income of employees for upcoming year so we have to interpret the income of employees on the basis of previous year's income of those employees
Saeed
thank you saeed, Akash. I understood.
Bhavani
Finding correlation and regression
explain statistics whether it is a science or arts or both
I would say art is a creation. A chef is an artist. They create new dishes just like the painters. I believe one who creates something new, is an artist. So, Statistics is also an art, if you know it, you can create some new formula, theory, law, etcetera. It is also Science. So yes, it is both.
Rohan
how do you use the normal distribution table when testing the hypothesis
Davia
Rohan
percentages of all the possible outcomes are measured. This is so simple and bases on the questionnaire or interview schedule. It's just measuring the probability chances of high %age of the either part of the hypothesis ... dependent ..independent. data is classified on the basis of respondents
saifuddin
what percent of the students would be expected to score above 95?
inferential statistics is what?
in which we make infrences (hypothsis)
surpose a data set of 2,3,5,6,1,4 are given find median
lucy
Mean (average) 4... Median (middle term) 3.5.. Mode (frequency) every element in a set has 1 frequrncy
Akash
i arrange the data set in ascending order. that is, 1,2,3,4,5,6. then find the data set that falls in the middle. in this case, 3 & 4 fall in the middle. you then sum and obtain the average. that is, (3+4)/2=3.5. therefore, 3.5 is the median.
Gbenga
both of you are correct.
Joseph
hello guys
Abasikponke
thanks
lucy
great to be here
King
how does a line graph look
King
hi
Davia
hello
lucy
pls who knows how line graph look like
King
line graph usually have a straight line running through axis
Dike
am new here anyone willing to orient me?
Timothy
find the media of the following numbers 61,64,67,70,73
my body pls
lucy
67
Benmike
Benmike
what is the percentile for the set of data in the class C and frequency F(c,f)given by (9.3-9.7,2) (9.8-10.2,5) (10.3-10.7,12) (10.8-11.2,17) (11.3-11.7,14) (11.8-12.2,6) (12.3-12.7,3) (12.8-13.2,1)
how to find median
arrange ascending and desending order than the mid value is Median
rajendra
ok
Hrishe
what if it is a group data
Oloyede
mean/ medium/ mode
Michelle
n\2 and n+1\2
An operational manager at a manufacturing company is interested in the level of satisfaction of computer buyers. The manager has developed a satisfaction scale of 1-10 to mark their level of understanding with the company.What is the population of the interest?
Any clues
Virtual
how to use grouped and ungrouped data
Just a test from gplay
how come 5.67
by dividing 11.37 on 2
saifuddin
by dividing 11.34 on 2
saifuddin
what is index number?
vinayak
What is the differences between quota an lottery system of sampling
EGBE
What are the are the characteristics that are critically expedients in selecting the sample size
EGBE
fit a binomial distribution for the following data and test the goodness of fit x: 0 1 2 3 4 5 6 f: 5 18 28 12 7 6 4
solution
Mano
Mano
Simonsakala
It is a square chi
Nelson
But can't be a binomial because, the x numbers are 0 to 6, instead those would be "0" or "1" in a straight way
Nelson
You can do a chi-square test, but the assumption has to be a normal distribution, and the last f's number need to be "64"
Nelson
sorry the last f's numbers : "6 and 4" which are the observed values for 5 and 6 (expected values)
Nelson
hi
rajendra
can't understand basic of statistics ..
rajendra
Sorry I see my mistake, we have to calculate the expected values
Nelson
So we need this equation: P= (X=x)=(n to x) p^x(1-p)^n-x
Nelson
why it is not possible brother
ibrar
were n= 2 ( binomial) x= number of makes (0 to 6) and p= probability, could be 0.8.
Nelson
so after we calculate the expected values for each observed value (f) we do the chi-square. x^2=summatory(observed-expected)^2 / expected and compare with x^2 in table with 0.8
Nelson
tomorrow I'll post the answer, I'm so tired today, sorry for my mistake in the first messages.
Nelson
It is possible, sorry for my mistake
Nelson
two trader shared investment and buoght Cattle.Mr.Omer bought 255 cows & rented the farm for a period of 32 days. Mr. Ahmed grazed his Cattle for 25 days. Mr. Ahmed's cattle was 180 cows.Together they profited $7800. the rent of the farm is$ 3000 so divide the profit per gows/day for grazing day
Mohamed
how to start this book, who is reading thins first time
It is my first time reading this book
Good one
ihsan