# Z-Hypothesis Test and a Diversification Challenge

This year was tough. Very tough. Apart from the new skills, I earned myself a good bonus. Usually, I just go on a trip using this extra cash, but it’s been a while since I wanted to start my own business. So, I set up a small fashion store in my garage operating only on the weekends.

I ask my friends (who also happen to be very famous – read fictitious) to rate my collections. They rate from 1 star to five stars and my collections get around 3.0 stars on average with a standard deviation of 1.0.

This month, I collected clothes from five different suppliers and made it more diverse. I got the ratings for my new collection and took 100 sample ratings to check if this hypothesis is true for my new collection or has it increased.

Read our article on Hypothesis Testing Explained with Pizzas for getting an idea of hypothesis testing.

A Z hypothesis test is one of the most basic hypothesis tests. It is used when we have a large sample size (in our case 100) and when the standard deviation of the population is known (in our case 1). The population must be approximately a normal distribution and data must be independent of each other.

Using historical information and information from a sample we arrive a starting point. Further analysis can be planned based on the orientation of this starting point.

The null hypothesis is that the average rating is 3.0 stars.

The alternate hypothesis is that average rating is greater than 3.0 stars.

Z – Test Workbook

Average of 100 samples = 3.2

Z value = (Average of samples – Average) / (Standard Deviation/Square Root (Sample Size))

Z value = 2.2

In a normal distribution curve, 1 standard deviation represents 50% to the right, 2 standard deviations represent 75% to the right. Therefore, our value falls under greater than 75% to the right.

We can set our alpha level to 5% (region of rejection). The area for rejection region in the Z table is 1.645.

Comparing our calculated Z value and the Z value of the rejection region we can say the calculated Z value is greater. Therefore, we can reject the null hypothesis.

What if we change our alternate hypothesis?

The alternate hypothesis is that average rating is not equal to 3.0 stars. (It can be greater than or equal to 3.0)

In this case, we split the rejection region to 2.5% at the left tail of the curve and 2.5% towards the right tail of the curve.

1 – 0.025 = 0.975

Z score for 0.975 from Z table is 1.96.

Therefore, the rejection region is to the right of 1.96 and to the left of -1.96.

Z value = 2.2

We can reject the null hypothesis if calculated Z value is less than -1.96 or if it is greater than 1.96. Here, since 2.2 > 1.96, we can reject the null hypothesis.

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