Question

A hotel manager calculates that 12% of the hotel rooms are booked. If the manager is right, what is the probability that the proportion of rooms booked in a sample of 556 rooms would be less than 10%?

Answer 1

0.0735 = 7.35% probability that the proportion of rooms booked in a sample of 556 rooms would be less than 10%.

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

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
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

A hotel manager calculates that 12% of the hotel rooms are booked.

This means that
`p = 0.12`

Sample of 556 rooms

This means that
`n = 556`

Mean and standard deviation:

`\mu = p = 0.12`

`s = \sqrt{(p(1-p))/(n)} = \sqrt{(0.12*0.88)/(556)} = 0.0138`

What is the probability that the proportion of rooms booked in a sample of 556 rooms would be less than 10%?

This is the p-value of Z when X = 0.1. So

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (0.1 - 0.12)/(0.0138)`

`Z = -1.45`

`Z = -1.45` has a p-value of 0.0735

0.0735 = 7.35% probability that the proportion of rooms booked in a sample of 556 rooms would be less than 10%.