Welcome to a lesson
on one sample hypothesis testing.
A systematic way to make a decision
on whether to reject or not reject the null hypothesis
is to compare the p-value and a preset
or preconceived alpha called the significant level.
A p-value is the probability of getting a value
of the test statistic that is at least as extreme
as the test statistic from the sample.
This assumes the null hypothesis is true.
Below, we have the p-value illustrated
as area under the normal curve for a left-tailed test,
right-tailed test, and two-tailed test
where the test statistic can be a mean,
standard deviation, proportion, z-score, or T-score.
Notice how for the two-tailed test
if we are given the p-value to find each area,
we do have to multiply by one half or divide by two.
And now take a look at alpha.
Again, alpha is a level of significance,
which is also the probability, of a type one error,
which is the error of rejecting the null hypothesis
when the null hypothesis is true.
And once again, we have a left-tailed test,
right-tailed test, and two-tailed test
where now the test statistic is a z-score.
For example, notice how when alpha is equal to 0.05
the z-score is negative 1.645,
and the region to the left of the z-score
is called the rejection region
that does have an area of 0.05.
For the right-tailed test, we have alpha equals 0.025.
The z-score is 1.96.
The area shaded to the right of the z-score, again,
is the rejection region, which has an area equal to 0.025.
For the two-tailed test though,
to find the areas on the left and right,
we do have to divide alpha by two, 0.1 divided by two.
It does give us 0.05.
So if we were comparing the z-score for the sample data
to the level of significance
and the z-score of the sample data fell
in the rejection region,
we would reject the null hypothesis.
If the z-score of the sample data
was not in the rejection region,
we would not reject the null hypothesis,
but in our case, we're comparing alpha and the p-value
and when alpha is greater than or equal to the p-value
or when the p-value is less than or equal to alpha,
we reject the null hypothesis.
If alpha is less than the p-value
or the p-value is greater than alpha,
then we do not reject the null hypothesis.
In general, a large p-value calculated from the data
indicates that we should not reject the null hypothesis.
The smaller the p-value, the more unlikely the outcome
and the stronger the evidence is
against the null hypothesis.
We would reject the null hypothesis
if the evidence is strongly against it.
So again, to make a decision as to whether we reject
or not reject the null hypothesis, we do as follows.
If alpha is greater than or equal to the p-value
or the p-value is less than or equal to alpha,
we can reject the null hypothesis.
The results of the sample are significant.
There is sufficient evidence to conclude
the null hypothesis is an incorrect belief
and that the alternative hypothesis may be correct.
If alpha is less than the p-value
or the p-value is greater than alpha,
we do not reject the null hypothesis.
The results of the sample data are not significant.
There is not sufficient evidence to conclude
that the alternative hypothesis may be correct.
It's also important to remember
when we do not reject the null hypothesis
it does not mean that we should believe
the null hypothesis is true.
It simply means that the sample data
have failed to provide sufficient evidence
to cast serious doubt about the truthfulness
of the null hypothesis.
Once we compare alpha and the p-value,
we should make a decision and write a thoughtful conclusion.
When using the p-value to evaluate a hypothesis test,
it is sometimes useful to use the following memory device.
If the p-value is low, the null must go.
If the p-value is high, the null must fly.
This memory aid relates a p-value less than
the established alpha as rejecting the null hypothesis,
given by if the p-value is low, the null must go.
Likewise, relates a p-value higher
than the established alpha
as not rejecting the null hypothesis,
indicated by if the p-value is high, the null must fly.
Another way to state the following information
is here below in bold.
You may want to pause the video and read over this.
To decide whether the alternative hypothesis
is less than greater than or not equal,
we do want to pay attention to the key words shown here.
And to find the test statistic
which will be a z-score or T-score as well as the p-value,
we will be using the TI-84 and perform a z-test,
T-test, or 1-PropZTest.
To know which test to use,
we will follow the guidelines shown here
where if the experiment or survey question
is a yes or no question, we use a 1-PropZTest
and the test statistic is a z-score,
if the experiment or survey is quantitative
and if the population standard deviation is known,
we use the z-test and the test statistic is a z-score,
if the experiment or survey is quantitative
and if the population standard deviation is unknown,
we use a T-test and the test statistic is a T-score.
Remember, if you are given data,
we need to make sure we select data once in the test editor,
otherwise we will use the stats option.
Let's take a look at an example.
Jeffrey, as an eight year old, established a mean time
of 16.43 seconds for swimming in the 25-yard freestyle,
with a standard deviation of 0.8 seconds.
His dad, Frank, thought that Jeffrey could swim
the 25-yard freestyle faster using goggles.
Frank bought Jeffrey a new pair of expensive goggles
and timed Jeffrey for 15 25-yard freestyle swims.
For the 15 swims, Jeffrey's mean time was 16 seconds.
Frank thought that the goggles helped Jeffrey
to swim faster than the 16.43 seconds,
which is the established mean.
Conduct a hypothesis test using a preset alpha equals 0.05.
Assume the swim times for the 25-yard freestyle are normal.
Let's first list the given information.
The established mean is 16.43 seconds,
and therefore our mu equals 16.43.
The standard deviation is given as 0.8 seconds.
Sigma equals 0.8.
Jeffrey swam 15 times, and therefore n,
the sample size, is 15.
And for the sample swims, the mean was 16 seconds.
X-bar equals 16.
Since Frank thought the goggles helped Jeffery swim faster
than the 16.43 seconds, we can represent this
by mu is less than 16.43.
And if mu is not less than 16.43,
we assume mu is equal to the established mean of 16.43.
So we have the null hypothesis, mu equals 16.43,
and we have the alternative hypothesis,
mu is less than 16.43.
Because we know the standard deviation,
we know the test statistic is a z-score,
and then looking at the graph,
we know the population mean is 16.43,
or the established mean, and then the sample mean is 16.
The array to the left of 16 under the curve
represents the p-value.
The p-value is equal to the probability X-bar,
the sample mean, is less than or equal to 16.
So now we need to go to the TI-84
and determine the z-score and the p-value.
To do this, we press stat, right arrow to tests.
We want to perform a z-test, so we press enter.
We're using the stats option under input,
which I've already selected.
Arrow down.
I've already entered the key information.
Mu is 16.43, enter.
Sigma is .8, enter.
X-bar is 16, enter.
N is 15, enter.
Now we need to be careful here.
Remember, the alternative hypothesis
is this mu is less than 16.43,
and therefore we select the less than mu,
which is already selected in the middle.
If it wasn't, we would highlight it,
press enter, go down to calculate, and press enter.
And we have the z-score, and we have the p-value.
Z is approximately negative 2.0817,
and the p-value is approximately 0.0187.
Notice a z-score is in the rejection region
where an alpha is equal to 0.05
and we have a left-tailed test.
Notice a z-score of negative 2.0817
is to the left of the z-score negative 1.645.
So we are going to reject the null hypothesis,
but before we do this formally, let's interpret the p-value
and then compare the p-value to alpha.
Remember, mu is equal to 16.43,
which comes from the null hypothesis,
and therefore, the assumption is mu equals 16.43,
which means the interpretation of the p-value
is if the null hypothesis is true,
then there is a 0.0187 probability, or 1.87% chance,
that Jeffrey's mean time to swim the 25-yard freestyle
is 16 seconds or less.
Because a 1.87% chance is small,
the mean time of 16 seconds or less
is unlikely to have happened randomly.
It is a rare event.
If we compare the alpha and the p-value,
notice how alpha is greater than
or equal to the p-value, where the p-value
is less than or equal to alpha,
and therefore we reject the null hypothesis.
This means we reject mu equals 16.43.
In other words, we do not think Jeffery swims
the 25-yard freestyle in 16.43 seconds
but faster with the new goggles.
At a 5% significance level,
we conclude that Jeffrey swims faster using the new goggles.
The sample data show there is sufficient evidence
that Jeffery's mean time to swim the 25-yard freestyle
is less than 16.43 seconds.
I hope you found this helpful.