Calculating a Two-Sample Z Test by Hand

Here is the problem you just saw in a previous presentation. Follow along doing the calculations on your own problem as you view this one.

Here is the problem you just saw in a previous presentation. Follow along doing the calculations on your own problem.

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

What are the researchers asking here?

What are the researchers asking here?

They are trying to determine if there is a difference in reported anxiety symptoms between those taking the new anti-anxiety drug and those taking the placebo.

What are the researchers asking here?

They are trying to determine if there is a difference in reported anxiety symptoms between those taking the new anti-anxiety drug and those taking the placebo.

What makes this question one that can be answered using a two-sample Z test is the fact that we are examining the difference between proportions (64 out of 200 or .32 compared to 92 out 200 or .46).

What are the researchers asking here?

They are trying to determine if there is a difference in reported anxiety symptoms between those taking the new anti-anxiety drug and those taking the placebo.

What makes this question one that can be answered using a two-sample Z test is the fact that we are examining the difference between proportions (64 out of 200 or .32 compared to 92 out 200 or .46).

Are these differences similar enough to make any differences we would find if we were to repeat the experiment to be due to chance or not?

With that in mind, we are able to state the null-hypothesis and then determine if we will reject or fail to reject it:

With that in mind, we are able to state the null-hypothesis and then determine if we will reject or fail to reject it:

There is NO significant proportional difference in reported anxiety symptoms between a sample of

participants who took a new anti-anxiety drug and a sample who took a placebo.

Since the test uses a .05 alpha value, that is what we will use to determine if the probability of a meaningful difference is rare or common.

Since the test uses a .05 alpha value, that is what we will use to determine if the probability of a meaningful difference is rare or common.

The alpha value makes it possible to determine what is called the z critical. If the z statistic that we are about to calculate from the data in the question is outside of the z critical [for a one-tailed test (e.g., +1.64) or a two-tailed test (e.g., -1.96 or +1.96]

One tailed test visual depiction:

rarecommon

+1.64

If the z value we are about to

calculate lands above this point, we will reject the null hypothesis

One tailed test visual depiction:

rarecommon

+1.64

If the z value lands below this point

we will fail to reject the null

hypothesis

A one tailed test could also go the other direction if we are testing the probability of one sample having a smaller proportion than another.

rare common

-1.64

A two-tailed test implies that we are not sure as to which direction it will go. We don’t know if the placebo or the new anxiety medicine will have better results.

Here is a visual depiction of the two-tailed test.

rare

-1.96

Common rare

+1.96

All we have left to do now is calculate the z statistic.

All we have left to do now is calculate the z statistic.Here is the formula for the z statistic for a Two-Sample Z-Test:

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(�̂�1− �̂�2)

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2

In the numerator we are subtracting one sample proportion from another sample proportion:

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(�̂�1− �̂�2)

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2

In the numerator we are subtracting one sample proportion from another sample proportion:

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(�̂�1− �̂�2)

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers want to test the effectiveness of a new anti-

anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

In the numerator we are subtracting one sample proportion from another sample proportion:

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(�̂�1− �̂�2)

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers want to test the effectiveness of a new anti-

anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

In the numerator we are subtracting one sample proportion from another sample proportion:

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(�̂�1− �̂�2)

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers want to test the effectiveness of a new anti-

anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

92/200=.4664/200=.32

In the numerator we are subtracting one sample proportion from another sample proportion:

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=( .32− .46 )

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers want to test the effectiveness of a new anti-

anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

92/200=.4664/200=.32

In the numerator we are subtracting one sample proportion from another sample proportion:

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(−.𝟏𝟐)

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers want to test the effectiveness of a new anti-

anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

The denominator is the estimated standard error:

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(− .12)

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2Researchers want to test the effectiveness of a new anti-

anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

This proportion is called the pooled

standard deviation. It is the same value

we use with independent

sample t-tests.

The denominator is the estimated standard error:

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(− .12)

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2

It is interesting to note that when you multiply a proportion (.80) by its complement (1-.8 = .20)

you get the variance (.80*.20 = .16).If you square that amount it comes to .04 and that

is the standard deviation.

Why is this important? Because the standard deviation divided by the square root of the sample size is the standard error.

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(− .12)

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2

It is interesting to note that when you multiply a proportion (.80) by its complement (1-.8 = .20)

you get the variance (.80*.20 = .16).If you square that amount it comes to .04 and that

is the standard deviation.

And the larger the standard error the less likely the two groups will be statistically significantly different.

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(− .12)

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2

It is interesting to note that when you multiply a proportion (.80) by its complement (1-.8 = .20)

you get the variance (.80*.20 = .16).If you square that amount it comes to .04 and that

is the standard deviation.

Conversely, the smaller the standard error the more likely the two groups will be statistically significantly different.

𝒁 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄=(− .12)

√�̂� (1−�̂� ) √ 1𝑛1+ 1𝑛2

It is interesting to note that when you multiply a proportion (.80) by its complement (1-.8 = .20)

you get the variance (.80*.20 = .16).If you square that amount it comes to .04 and that

is the standard deviation.

So, let’s compute the pooled standard deviation:

So, let’s compute the pooled standard deviation:

�̂�=𝑥1+𝑥2𝑛1+𝑛2

So, let’s compute the pooled standard deviation:

�̂�=𝑥1+𝑥2𝑛1+𝑛2

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

Here’s the problem again:

So, let’s compute the pooled standard deviation:

�̂�=𝑥1+𝑥2𝑛1+𝑛2

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

So, let’s compute the pooled standard deviation:

�̂�=64+𝑥2𝑛1+𝑛2

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

So, let’s compute the pooled standard deviation:

�̂�=64+𝑥2𝑛1+𝑛2

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

So, let’s compute the pooled standard deviation:

�̂�=64+𝑥2200+𝑛2

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

So, let’s compute the pooled standard deviation:

�̂�=64+𝑥2200+𝑛2

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

So, let’s compute the pooled standard deviation:

�̂�=64+92200+𝑛2

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

So, let’s compute the pooled standard deviation:

�̂�=64+92200+𝑛2

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

So, let’s compute the pooled standard deviation:

�̂�=64+92200+200

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

Add the fractions:

�̂�=64+92200+200

Add the fractions:

�̂�=156

200+200

Add the fractions:

�̂�=156

200+200

Add the fractions:

�̂�=156200

Add the fractions:

�̂�=.78

Now we plug this pooled proportion:

�̂�=.78

Now we plug this pooled proportion:

into the standard error formula in the denominator

�̂�=.78

Now we plug this pooled proportion:

into the standard error formula in the denominator

�̂�=.78

Now we plug this pooled proportion:

into the standard error formula in the denominator

�̂�=.78

Now we plug this pooled proportion:

into the standard error formula in the denominator

Now we plug this pooled proportion:

into the standard error formula in the denominator

Now we plug this pooled proportion:

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒)

√ .𝟏𝟕𝟐√ 1𝒏𝟏+ 1𝑛2

Now we need to calculate n or the sample size:

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒)

√ .172√ 1200

+ 1𝑛

Now we need to calculate n or the sample size:

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒)

√ .172√ 1200

+ 1200

Researchers want to test the effectiveness of a new anti-anxiety medication. In clinical testing, 64 out of 200 people taking the medication report symptoms of anxiety. Of the

people receiving a placebo, 92 out of 200 report symptoms of anxiety. Is the medication working any differently than the

placebo? Test this claim using alpha = 0.05.

𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒)

√ .172√ 1200

+ 1200

Now we need to calculate the Zstatistic

𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒 )

√ .172√ 2200

Now we need to calculate the Zstatistic

𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒 )

√ .172√ 1100

Now we need to calculate the Zstatistic

𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒)

√ .172√ .01

Now we need to calculate the Zstatistic

𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(− .𝟏𝟒 )√ .172(.1)

Now we need to calculate the Zstatistic

𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(− .𝟏𝟒 )

(.414)(.1)

Now we need to calculate the Zstatistic

𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=(−.𝟏𝟒).0414

Now we need to calculate the Zstatistic

Now we need to calculate the Zstatistic

𝑍 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐=−3 .379

Let’s see where this lies in the distribution:

rare

-1.96

Common rare

+1.96

Let’s see where this lies in the distribution:

rare

-1.96

Common rare

+1.96-3.38

Because -3.38 is outside the -1.96 range we consider it to be a rare occurrence and therefore we will reject the null hypothesis and accept the alternative hypothesis:

Because -3.38 is outside the -1.96 range we consider it to be a rare occurrence and therefore we will reject the null hypothesis and accept the alternative hypothesis:

There was a significant difference in effectiveness between the medication group and the placebo group, z = -3.379, p < 0.05.