Question

A hotel manager believes that 23% 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 610 rooms would differ from the population proportion by less than 3%

Answer 1

0.9216 = 92.16% probability that the proportion of rooms booked in a sample of 610 rooms would differ from the population proportion by less than 3%

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 believes that 23% of the hotel rooms are booked.

This means that
`p = 0.23`

Sample of 610 rooms

This means that
`n = 610`

Mean and standard deviation:

`\mu = p = 0.23`

`s = \sqrt{(p(1-p))/(n)} = \sqrt{(0.23*0.77)/(610)} = 0.017`

What is the probability that the proportion of rooms booked in a sample of 610 rooms would differ from the population proportion by less than 3%?

p-value of Z when X = 0.23 + 0.03 = 0.26 subtracted by the p-value of Z when X = 0.23 - 0.03 = 0.2. So

X = 0.26

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

By the Central Limit Theorem

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

`Z = (0.26 - 0.23)/(0.017)`

`Z = 1.76`

`Z = 1.76` has a p-value of 0.9608

X = 0.2

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

`Z = (0.2 - 0.23)/(0.017)`

`Z = -1.76`

`Z = -1.76` has a p-value of 0.0392

0.9608 - 0.0392 = 0.9216

0.9216 = 92.16% probability that the proportion of rooms booked in a sample of 610 rooms would differ from the population proportion by less than 3%