If you recall, a proportion is a part, or a number in relation to the whole. It is given as a percentage, for example 78% of all dogs bark when the doorbell rings, and this 78
can be written in decimal form as 0.78 or as a fraction, 78 over 100, which can be reduced to 39 over 50. And 78%, 0.78 and 39 over 50 are all equal to each other.
We can do hypothesis tests for proportions by using a z test. The formula we use to do this is z equals p hat, minus p, divided by square root of the quantity, p times q,
divided by n. And in this formula, p hat is the proportion of a sample, which is called the test statistic, p is the hypothesized population proportion, q is 1 minus p,
n is the sample size and z is the standardized test statistic. In order to use this formula we must verify that the sampling distribution for p hat can
be approximated by a normal distribution, so 2 conditions must be met, n times p must be greater than or equal to 5, and n times q must also be greater than or
equal to 5. And in doing hypothesis testing for proportions, we use similar concepts used for testing for means, such as, left tailed, right tailed and 2 tailed tests with the
level of significance, alpha being shaded areas that are rejection regions, and critical values representing the boundaries for the rejection regions. Lets go through an
example step by step for a hypothesis test for a proportion. A researcher claims that less than 54% of people text other people when they are dinning out. In a random
sample of 84 people that were dinning out, 35 of them texted other people during their dinner. At a level of significance of alpha equals 0.01, is there enough evidence
to support the researcher's claim? Step 1 is to make sure the conditions are met to use a z test. n times p is 84, the sample size, times 0.54, the population proportion,
which equals 45.36, which is greater than or equal to 5. And n times q is 84 times 0.46, which equals 38.64, which is also greater than or equal to 5. So all of
the conditions are met. Step 2 is to write the claim out and identify the null and alternative hypotheses. The claim is that p, the population proportion is less than 0.54.
The alternative hypothesis contains a statement of inequality, so h sub a is p less than 0.54. The null hypothesis is the complement of the alternative hypothesis and contains
a statement of equality, so h sub 0 is p is greater than or equal to 0.54. Step 3 is to identify the level of significance, which was given, alpha equals 0.01. Step 4,
is to determine the test to use, left tailed, right tailed or 2 tailed, and because the alternative hypothesis contains the less than inequality, this will be a left tailed test.
Step 5 is to determine the critical value or values and since this is a one tailed test, a left tailed test, there will be only 1 critical value. Since the level of significance,
is alpha equals 0.01, we need to find the value for z naught, the critical value, in a z distribution table where the area to the left equals 0.01. And in the table that value is
-2.33, so that is our critical value, z naught. Step 6 is to identify the rejection region, and our rejection region is any standardized test statistic value that falls in the shaded
area, that is any value that is less than z naught. Step 7 is to use the formula and calculate the z value, or the value of the standardized test statistic.
Calculating for p hat, we have 35 divided by 84, which equals approximately 0.417, and using our formula we have, 0.417 minus 0.54, divided by the square root of 0.54
times 0.46 divided by 84. Which calculates out to -2.268. Step 8 is to make a decision to reject or fail to reject the null hypothesis. On our graph, you can see that the
standardized test statistic does not fall in the rejection region, as z, the standardized test statistic is greater than z naught, the critical value. So, in this case we would fail to
reject the null hypothesis. Step 9 is to interpret the decision. There is not enough evidence at the 1% level of significance to support the claim that less than 54% of
people text other people when they are dinning out. One more example. A recent report claimed that 62% of people watch sports every day. In a random sample of 350
people, 53% of people said they watch sports every day. At a level of significance of alpha equals 0.10, is there enough evidence to support the claim? Step 1 is to make
sure the conditions are met to use a z test. n times p equals 217, which is greater than or equal to 5. And n times q equals 133, which is also greater than or equal to 5.
So all of the conditions are met. Step 2 is to write the claim out and identify the null and alternative hypotheses. The claim is that p, the population proportion equals 0.62.
The null hypothesis contains a statement of equality, so h sub 0 is p equals 0.62. And the alternative hypothesis contains a statement of inequality, so h sub a is p does not
equal 0.62. Step 3 is to identify the level of significance, which was given, alpha equals 0.10. Step 4, is to determine the test to use, left tailed, right tailed or 2 tailed,
and because the alternative hypothesis contains the does not equals inequality, this will be a 2 tailed test. Step 5 is to determine the critical value or values and since this
is a 2 tailed test, there will be 2 critical values. Graphically this looks like this, with our critical values, negative z naught and positive z naught, here and both of these shaded
areas in the tails being rejection regions. Since the level of significance, is alpha equals 0.10, and this is a 2 tailed test, each of these rejection regions will be one half
of alpha, which is 0.05. And in the z distribution table that value is -1.645 and positive 1.645, so those are our critical values. Step 6 is to identify the rejection regions,
and our rejection regions is any standardized test statistic value that falls in the shaded area, that is any value that is less than negative z naught, or any value that
is greater than positive z naught, which is any value less than -1.645 or any value greater than positive 1.645. Step 7, use the formula and calculate the z value,
or the value of the standardized test statistic. And using our formula we have, 0.55 minus 0.62, divided by the square root of 0.62, times 0.38, divided by 350.
Which calculates out to -2.698. Step 8 is to make a decision to reject or fail to reject the null hypothesis. On our graph, you can see that the standardized test statistic
does fall in the rejection region to the left, as z, the standardized test statistic is less than negative z naught, the critical value. So, in this case we would reject the null
hypothesis. Step 9 is to interpret the decision. There is enough evidence at the 10% level of significance to reject the claim that 62% of people watch sports every day.
Alright my friends, that be the basics on doing hypothesis tests for a population proportion. Hopefully this video helped you out,
I do have more videos right there for you, till next time, I am outta here.