Find the value of x in the following diagram.
1 answer:
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Answer:
14.69% probability that this happens
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use 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.
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.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean and standard deviation
1000 people were given assurance of a room.
This means that
Let us assume that each customer cancels their reservation with a probability of 0.1.
So 0.9 probability that they still keep their booking, which means that
Probability more than 900 still keeps their booking:
So
901/1000 = 0.91
So this is 1 subtracted by the pvalue of Z when X = 0.91.
By the Central Limit Theorem
has a pvalue of 0.8531
1 - 0.8531 = 0.1469
14.69% probability that this happens