Calidad

10.1 Hypothesis Testing for Two Population Means
Hypothesis Testing for Two Population Means, Independent Groups
The distribution for a hypothesis test of two population means, independent groups, is the Student-t when the population standard deviations of the two groups are unknown or the normal if the population standard deviations are known. Knowing the population standard deviations ishighly unlikely.
The statistic is the difference between the sample means of the two groups calculated from the data. If we call the groups group 1 and group 2, then the random variable is:
[pic]
and the statistic is:
[pic]
(read as "xbar1 minus xbar2")
where
[pic]
(read as "xbar1")
is the sample mean of the sample taken from group 1 and
[pic]
(read as "xbar2")
is the sample mean of thesample taken from group 2.
The two independent groups are populations that are normally distributed.
When the null hypothesis is that there is no difference between the population group means, we write Ho as
μ1 - μ2 = 0
Depending on the test, Ho may be written as
[pic]
or
[pic]
If Ho is μ1 - μ2 = 0, then we are testing μ1 = μ2.
For the other cases, we could write:
[pic]
and
[pic]
Backto Top
Hypothesis Testing Problems Using TI-83 or TI-84 calculators
Example:
Companies that produce the same type of product often compare their products. In one particular test, the number of M & M's (registered trademark) plain candies were compared to the number of Reeses Pieces (registered trademark). In a sample of 40 small packages of M & Ms, the sample mean was 57.7 candies with a samplestandard deviation of 2.1 candies. In a sample of 35 packages of the same size of Reeses Pieces, the sample mean was 52.6 candies with a sample standard deviation of 2.4 candies. The populations of the number of M & M's candies in a small package and the number of Reeses Pieces in the same size package are normal.
Conduct an appropriate hypothesis test to determine whether or not the averagenumber of candy pieces per package is the same for the two brands. Use a level of significance of 0.01.
The important numbers are:
• for the M & M's candies, sample mean = 57.7, sample standard deviation = 2.1, sample size = 40, and
• for the Reeses Pieces, sample mean = 52.6, sample standard deviation = 2.4, same size = 35.
Formulate the 2 hypotheses.
Let the M & Ms be group 1 and theReeses Pieces be group 2.
Ho: μ1 = μ2.
[pic]
two-tailed test
Determine the random variable and the distribution for the test.
[pic]
Student-t because the population standard deviations are unknown.
Using the statistic calculated from the data, calculate the p-value.
[pic]
Use TI-83 or TI-84 calculator functions.
STAT TESTS 4: 2-SampTTest.
Inpt: Stat
xbar1: 57.7
sx1: 2.1
n1: 40
xbar2:52.6
sx2: 2.4
n2: 35
μ1: =/= μ2
Pooled: No
(To pool would be to combine the sample standard deviations into one special standard deviation. We do not need to do this.)
Calculate
The p-value (p) is 1.625151E-14 which is almost 0. (Reminder: 1.625151E-14 is equal to 1.625151 multiplied by 10-14.)
Compare α and the p-value and make a decision.
α= 0.01
Since 0.01 > 0 ( α> p-value), we rejectHo.
Write an appropriate conclusion.
We conclude that the there is a difference between the average number of M & Ms and Reeses Pieces in small packages of the same size.
Example:
Two machines fill grated cheese packages. The population standard deviations are known to be 0.80 ounces for machine A and 0.60 ounces for machine B. Samples are selected from each machine. The sample data is asfollows:
|--- |Sample Size |Sample Mean |
|Machine A |32 |8.2 ounces |
|Machine B |34 |7.9 ounces |


We are interested in determining whether the mean content of packages filled by machine A is more than the mean content of packages filled by...