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A smart phone manufacturer claims that

less than 10% of its devices fail within one year.

A customer advocacy group tests 500 smartphones and

finds that 30 of them failed within the 12 months.

Does the data support the manufacturer's claim at 5% level of significance?

So we want to do this as a hypothesis test,

the claim is that less than will fail.

So as you can see less than does not have the equal to sign and

it doesn't say 10% or less.

It says less than 10%, so we have to put now the claim that

the manufacturer makes as the ultimate hypothesis.

So the ultimate hypothesis is that proportion level fail is less than 10%.

Therefore, our null is that the proportion would be greater than or equal to the 10%.

One of the things that I have not mention in my lessons is that when you're doing

a testing on proportion.

Proportion is not a continuous variable for example,

a smartphone either fails or it doesn't fail, so it's a zero, one.

It's a discreet variable, not a continuous variable.

But if you have a large enough sample, where n times p and

n times (1-p) is greater than equal to 5.

Then you can use normal distribution to approximate for

what is really known as a binomial distribution.

So I have not mentioned it because I have been saying to you that we will be using

large data sets.

Which in most cases to have a valid study you would need a large data set.

And remember large is not that big of a sample size.

Anything above 30, most of them is considered large enough.

So if these two conditions hold, we will use normal distribution.

Which means we would use the z value as measure of how many standard

errors away our samples from what we thought it should be.

So now that we have this, we can go ahead and calculate it.

And what we are looking for is this value of z and p-value.

Now z is calculated by taking the p hat, which is the sample proportion minus p0,

which is the hypothesized proportion.

And dividing that by the standard error, which is just p0 times

1 minus P0 divided by n when you take the entire square root.

So I'm going to use the right hand side to find these values.

So sample proportion, which is your p hat, is 38 out of the 500.

So in our sample, 7.6 failed, but we don't know if this is enough to support

the manufacturer's claim or not, so we're going to check that.

The hypothesis proportion is 10%, and

in this case is 500 cell phones that were tested or

smart phones that were tested level of significance is 0.05.

So now z is going to be and I'm going to use this formula to do the calculation.

So z is and I'm going to use parenthesis to not make any errors here.

So p hat it is right minus hypothesised proportion, so

that's my numerator divided by the square root of p0 which

is hypothesised proportion t imes 1- p0.

And dividing this value by the sample size of 500,

closing the parentheses, and you get the value of -1.7889.

So now what are we looking at?

We are looking at samples that will be normally distributed and

you would expect that if this is the true proportion of failure.

We would get this as the mean of the sampling distribution and

the sampling distribution would look normal.

And the standard error which is right here,

this denominator standard error is going to give me the shape.

Now, it's a left tail because we have a less than sign here, so it's a left tail.

So anything that deviates more than this 5% chance of finding,

then it will say that it will fall into a rejection region, we will reject it.

So if we deviate too much, you will end up rejecting.

That's a small p.

If the sample that you have taken is so rare,

you should not assume that you're just unlucky.

The first assumption should be that probably the claim,

the null hypothesis, is not true.

So now we have to find the p-value here.

So to find the p-value we will use a normal distribution, which is NORM.S.DIST.

That means that I'm going to give it a z value and then say one and see what I get.

I get the p-value of 0.03.

So what does that mean?

P-value is less than alpha therefore, we will reject the null hypothesis.

If you re inject the null hypothesis,

it means we must accept the claim that less than 10% of the smartphones from this

manufacturer will fail within the first year.

1-5.2. Left-Tail Test for Proportion in Excel

Inferential and Predictive Statistics for Business

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