Hi! ⋇
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All proportions have <u>this</u> <u>form</u>:
, Where
is equal to
.
If , it's <u>not</u> a proportion.
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Here we have two pairs of numbers:
40,10 and 32,8.
Written As a <u>proportion</u>, they <em>look like</em> :
Hope this made sense to you :)
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Answer:
Yes, it would be unusual.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
If or
, the outcome X is considered unusual.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this question, we have that:
Would it be unusual for this sample mean to be less than 19 days?
We have to find Z when X = 19. So
By the Central Limit Theorem
, so yes, the sample mean being less than 19 days would be considered an unusual outcome.