Assumptions for Z-score and t-score | Midterm Exam

Now that you've learned how to calculate z-scores and t-scores, let's learn the necessary assumptions to check before you calculate them in the future.

You might be wondering... why didn't we learn this before learning how to calculate z-scores and t-scores?

Frankly, it's just easier to first understand how to calculate them, then understand the assumptions you need to validate before calculating. Learning the assumptions first typically leads to students wondering why they even exist in the first place.

Assumptions allow us to ensure a sample has a normal distribution, enabling us to utilize all the formulas necessary to compute the probability of an event occurring.

To be clear, the distributions utilized for Z-scores and t-scores are both normal. The Z-score distribution is just referred to as the "standard" normal deviation.

With that being said, let's learn the assumptions needed when calculating z-scores / t-scores for a mean vs. proportion!

Assumptions with means

For problems involving a sample means, we have to check the following 3 assumptions:

1. Sample is randomly selected from the population
2. The sample size (n) is less than or equal to 10% of of the population size (N)
3. The sample size is greater than or equal to 30, or the population itself is normally distributed

If our sample does not pass these assumptions, then we cannot proceed with calculating the z-score or t-score.

Let's practice assessing these assumptions with a quick example:

Question: Amazon ships millions of packages a year using delivery trucks. They want to know the average weight of all packages they ship so that they can determine the maximum capacity of their trucks. They weigh 1000 randomly selected Amazon packages and determine that the mean weight is 2.5 pounds.

1.Sample is randomly selected from the population

The problem states that the sample was randomly selected from the population...

Question: Amazon ships millions of packages a year using delivery trucks. They want to know the average weight of all packages they ship so that they can determine the maximum capacity of their trucks. They weigh 1000 randomly selected Amazon packages and determine that the mean weight is 2.5 pounds.

...so this assumption is checked. 

2. The sample size (n) is less than or equal to 10% of of the population size (N)

The problem states that Amazon ships millions of packages a year...

Question: Amazon ships millions of packages a year using delivery trucks. They want to know the average weight of all packages they ship so that they can determine the maximum capacity of their trucks. They weigh 1000 randomly selected Amazon packages and determine that the mean weight is 2.5 pounds.

...and 1000 packages is less than or equal to 10% of all the packages that Amazon ships (which is the population size)...

Question: Amazon ships millions of packages a year using delivery trucks. They want to know the average weight of all packages they ship so that they can determine the maximum capacity of their trucks. They weigh 1000 randomly selected Amazon packages and determine that the mean weight is 2.5 pounds.

...so this assumption is checked. 

3. The sample size is greater than or equal to 30, or the population itself is normally distributed

The sample size is 1000...

Question: Amazon ships millions of packages a year using delivery trucks. They want to know the average weight of all packages they ship so that they can determine the maximum capacity of their trucks. They weigh 1000 randomly selected Amazon packages and determine that the mean weight is 2.5 pounds.

...which is greater than 30. Therefore, this assumption is checked.

Assumptions with proportions

For problems involving sample proportions, we have to check the following 3 assumptions:

1. Sample is randomly selected from the population
2. The sample size (n) is less than or equal to 10% of of the population size N
3. There are 10 successes and 10 failures in the sample OR np >= 10 and nq >= 10

If our sample does not pass these assumptions, then we cannot proceed with calculating the z-score (remember, you won't calculate t-scores with sample proportions).

np >= 10 represents the number of successes. nq >= 10 represents the number of failures... which can be calculated by doing the following formula:

q = 1 - p

q is essentially the opposite of p.

For example, if we had a proportion of 0.25 of successes...

q = 1 - 0.25

...that'd mean we'd have 0.75 failures.

q = 1 - 0.25 = 0.75

Let's practice assessing these assumptions with a quick example:

Crazy Christmas is a company that sells Christmas ornaments and decorations. They want to know whether 85% or more of US households celebrate Christmas. They conduct a survey with 1000 randomly selected participants and find that 750 of them do celebrate Christmas.

1. Sample is randomly selected from the population

The problem states that the sample was randomly selected from the population...

Crazy Christmas is a company that sells Christmas ornaments and decorations. They want to know whether 85% or more of US households celebrate Christmas. They conduct a survey with 1000 randomly selected participants and find that 750 of them do celebrate Christmas.

...so this assumption is checked. 

2. The sample size (n) is less than or equal to 10% of of the population size (N)

There are millions of households in the US (which is our population), so 1000 households is less than 10% of all US households.

Crazy Christmas is a company that sells Christmas ornaments and decorations. They want to know whether 85% or more of US households celebrate Christmas. They conduct a survey with 1000 randomly selected participants and find that 750 of them do celebrate Christmas.

Therefore, this assumption is checked.

3. There are 10 successes and 10 failures in the sample OR np >= 10 and n(1-p) >= 10

A “success” in this example would be a household that celebrates Christmas and a “failure” would be one that does not. This example has at least 10 successes (houses that celebrate Christmas)...

Crazy Christmas is a company that sells Christmas ornaments and decorations. They want to know whether 85% or more of US households celebrate Christmas. They conduct a survey with 1000 randomly selected participants and find that 750 of them do celebrate Christmas.

...and at least 10 failures (houses that don't celebrate Christmas). How many failures did we have? Based on the following equation...

# of failures = total respondents - # of successes

...we had 1000 total respondents...

# of failures = 1000 - # of successes

...and 750 of those were successes...

# of failures = 1000 - 750

...therefore we had 250 failures!

# of failures = 250

Based on this finding, we can affirm this assumption is checked.

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