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  1. The two independent samples are simple random samples from two distinct populations.
  2. For the two distinct populations:
    • if the sample sizes are small, the distributions are important (should be normal)
    • if the sample sizes are large, the distributions are not important (need not be normal)

The test comparing two independent population means with unknown and possibly unequal population standard deviations is called the Aspin-Welch t-test. The degrees of freedom formula was developed by Aspin-Welch.

The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. In order to account for the variation, we take the difference of the sample means, X ¯ 1 X ¯ 2 , and divide by the standard error in order to standardize the difference. The result is a t-score test statistic.

Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error , of the difference in sample means , X ¯ 1 X ¯ 2 .

The standard error is:

( s 1 ) 2 n 1 + ( s 2 ) 2 n 2

The test statistic ( t -score) is calculated as follows:

( x ¯ 1 x ¯ 2 ) ( μ 1 μ 2 ) ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2


  • s 1 and s 2 , the sample standard deviations, are estimates of σ 1 and σ 2 , respectively.
  • σ 1 and σ 1 are the unknown population standard deviations.
  • x ¯ 1 and x ¯ 2 are the sample means. μ 1 and μ 2 are the population means.

The number of degrees of freedom ( df ) requires a somewhat complicated calculation. However, a computer or calculator calculates it easily. The df are not always a whole number. The test statistic calculated previously is approximated by the Student's t -distribution with df as follows:

Degrees of freedom

d f = ( ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2 ) 2 ( 1 n 1 1 ) ( ( s 1 ) 2 n 1 ) 2 + ( 1 n 2 1 ) ( ( s 2 ) 2 n 2 ) 2

When both sample sizes n 1 and n 2 are five or larger, the Student's t approximation is very good. Notice that the sample variances ( s 1 ) 2 and ( s 2 ) 2 are not pooled. (If the question comes up, do not pool the variances.)

It is not necessary to compute this by hand. A calculator or computer easily computes it.

Independent groups

The average amount of time boys and girls aged seven to 11 spend playing sports each day is believed to be the same. A study is done and data are collected, resulting in the data in [link] . Each populations has a normal distribution.

Sample Size Average Number of Hours Playing Sports Per Day Sample Standard Deviation
Girls 9 2 0.866
Boys 16 3.2 1.00

Is there a difference in the mean amount of time boys and girls aged seven to 11 play sports each day? Test at the 5% level of significance.

The population standard deviations are not known. Let g be the subscript for girls and b be the subscript for boys. Then, μ g is the population mean for girls and μ b is the population mean for boys. This is a test of two independent groups , two population means .

Random variable : X ¯ g X ¯ b = difference in the sample mean amount of time girls and boys play sports each day.
H 0 : μ g = μ b    H 0 : μ g μ b = 0
H a : μ g μ b    H a : μ g μ b ≠ 0
The words "the same" tell you H 0 has an "=". Since there are no other words to indicate H a , assume it says "is different." This is a two-tailed test.

Distribution for the test: Use t df where df is calculated using the df formula for independent groups, two population means. Using a calculator, df is approximately 18.8462. Do not pool the variances.

Calculate the p -value using a Student's t -distribution: p -value = 0.0054


This is a normal distribution curve representing the difference in the average amount of time girls and boys play sports all day. The mean is equal to zero, and the values -1.2, 0, and 1.2 are labeled on the horizontal axis. Two vertical lines extend from -1.2 and 1.2 to the curve. The region to the left of x = -1.2 and the region to the right of x = 1.2 are shaded to represent the p-value. The area of each region is 0.0028.

s g = 0.866
s b = 1
So, x ¯ g x ¯ b = 2 – 3.2 = –1.2
Half the p -value is below –1.2 and half is above 1.2.

Make a decision: Since α > p -value, reject H 0 . This means you reject μ g = μ b . The means are different.

Press STAT . Arrow over to TESTS and press 4:2-SampTTest . Arrow over to Stats and press ENTER . Arrow down and enter 2 for the first sample mean, 0.866 for Sx1, 9 for n1, 3.2 for the second sample mean, 1 for Sx2, and 16 for n2. Arrow down to μ1: and arrow to does not equal μ2. Press ENTER . Arrow down to Pooled: and No . Press ENTER . Arrow down to Calculate and press ENTER . The p -value is p = 0.0054, the dfs are approximately 18.8462, and the test statistic is -3.14. Do the procedure again but instead of Calculate do Draw.

Conclusion: At the 5% level of significance, the sample data show there is sufficient evidence to conclude that the mean number of hours that girls and boys aged seven to 11 play sports per day is different (mean number of hours boys aged seven to 11 play sports per day is greater than the mean number of hours played by girls OR the mean number of hours girls aged seven to 11 play sports per day is greater than the mean number of hours played by boys).

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Questions & Answers

what is standard deviation?
Jawed Reply
It is the measure of the variation of certain values from the Mean (Center) of a frequency distribution of sample values for a particular Variable.
Yeah....the simplest one
what is the number of x
Godgift Reply
Javed Arif
how will you know if a group of data set is a sample or population
Kingsley Reply
population is the whole set and the sample is the subset of population.
if the data set is drawn out of a larger set it is a sample and if it is itself the whole complete set it can be treated as population.
hello everyone if I have the data set which contains measurements of each part during 10 years, may I say that it's the population or it's still a sample because it doesn't contain my measurements in the future? thanks
Pls I hv a problem on t test is there anyone who can help?
What's your problem Peggy Abang
Bhavika is right
what is the problem peggy?
hii Bhavika
Hi eny population has a special definition. if that data set had all of characteristics of definition, that is population. otherwise that is a sample
three coins are tossed. find the probability of no head
Kanwal Reply
three coins are tossed consecutively or what ?
p(getting no head)=1/8
or .125 is the probability of getting no head when 3 coins are tossed
what is two tailed test
Umar Reply
if the diameter will be greater than 3 cm then the bullet will not fit in the barrel of the gun so you are bothered for both the sides.
in this test you are worried on both the ends
lets say you are designing a bullet for thw gun od diameter equals 3cm.if the diameter of the bullet is less than 3 cm then you wont be able to shoot it
In order to apply weddles rule for numerical integration what is minimum number of ordinates
Anjali Reply
excuse me?
didn't understand the question though.
which question? ?
We have rules of numerical integration like Trapezoidal rule, Simpson's 1/3 and 3/8 rules, Boole's rule and Weddle rule for n =1,2,3,4 and 6 but for n=5?
geometric mean of two numbers 4 and 16 is:
iphone Reply
quartile deviation of 8 8 8 is:
sorry 8 is the geometric mean of 4,16
quartile deviation of 8 8 8 is
can you please expalin the whole question ?
can you please post the picture of that ?
10 now
how to find out the value
srijth Reply
can you be more specific ?
what is the difference between inferential and descriptive statistics
Eze Reply
descriptive statistics gives you the result on the the data like you can calculate various things like variance,mean,median etc. however, inferential stats is involved in prediction of future trends using the previous stored data.
if you need more help i am up for the help.
Thanks a lot
Inferential Statistics involves drawing conclusions on a population based on analysis of a sample. Descriptive statistics summarises or describes your current data as numerical calculations or graphs.
my pleasure😊. Helping others offers me satisfaction 😊
for poisson distribution mean............variance.
mehul Reply
both are equal to mu
what is a variable
Bonolo Reply
something that changes
why we only calculate 4 moment of mean? asked in papers.
Faizan Reply
why we only 4 moment of mean ? asked in BA exam
Good evening, can you please help me by sharing regression and correlation analysis notes....thank you in advance
Refiloe Reply
Hello, can you please share the possible questions that are likely to be examined under the topic: regression and correlation analysis.
for normal distribution mean is 2 & variance is 4 find mu 4?
Faizan Reply
repeat quastion again
find mu 4. it can be wrong but want to prove how.
for a normal distribution if mu 4 is 12 then find mu 3?
Faizan Reply
Question hi wrong ha
ye BA mcqs me aya he teen he. 2dafa aya he
if X is normally distributed. (n,b). then its mean deviation is?
The answer is zero, because all odd ordered central moments of a normal distribution are Zero.
which question is zero
sorry it is (5,16) in place of (n,b)
I got. thanks. it is zero.
a random variable having binomial distribution is?

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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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