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alex41 [277]
3 years ago
6

A hotel manager believes that 27% of the hotel rooms are booked. If the manager is correct, what is the probability that the pro

portion of rooms booked in a sample of 423 rooms would differ from the population proportion by less than 6%
Mathematics
1 answer:
AlladinOne [14]3 years ago
6 0

Answer:

The probability that the proportion of rooms booked in a sample of 423 rooms would differ from the population proportion by less than 6% is 0.9946.

Step-by-step explanation:

According to the Central limit theorem, if from an unknown population large samples of sizes n > 30, are selected and the sample proportion for each sample is computed then the sampling distribution of sample proportion follows a Normal distribution.

The mean of this sampling distribution of sample proportion is:

 \mu_{\hat p}=p

The standard deviation of this sampling distribution of sample proportion is:

 \sigma_{\hat p}=\sqrt{\frac{\hat p(1-\hat p)}{n}}

The information provided here is:

<em>p</em> = 0.27

<em>n</em> = 423

As <em>n </em>= 423 > 30, the sampling distribution of sample proportion can be approximated by the Normal distribution.

The mean and standard deviation of the sampling distribution of sample proportion are:

\mu_{\hat p}=p=0.27\\\\\sigma_{\hat p}=\sqrt{\frac{\hat p(1-\hat p)}{n}}=\sqrt{\frac{0.27\times(1-0.27)}{423}}=0.0216

Compute the probability that the proportion of rooms booked in a sample of 423 rooms would differ from the population proportion by less than 6% as follows:

P(|\hat p-p|

                           =P(0.27-0.06

*Use a <em>z</em>-table.

Thus, the probability that the proportion of rooms booked in a sample of 423 rooms would differ from the population proportion by less than 6% is 0.9946.

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