Question

An article claimed that only 38% of hotel visitors used the alarm clock provided. A hotel manager wanted to see whether that proportion applied to the hotel where she worked, so she took a random sample of 700 of the approximately 10,000 visitors to the hotel one month and checked whether they used their room alarm clock. Of the sampled visitors, only 35% used the alarm clock. Assuming that the article's 38% claim is also the true proportion for visitors to the hotel, what is the approximate probability that fewer than 35% of the sample used the alarm clock?

Answer 1

To find the approximate probability that fewer than 35% of the sample used the alarm clock, we can use a normal distribution. First, calculate the standard deviation of the sample proportion. Then, find the z-score using the sample proportion, population proportion, and standard deviation. Finally, use a standard normal distribution table or calculator to find the probability.

Step-by-step explanation:

To find the approximate probability that fewer than 35% of the sample used the alarm clock, we need to use a normal distribution since the sample size is large enough. We know that the article claimed that 38% of hotel visitors used the alarm clock, and we are assuming this claim is true. We also know that in our sample of 700 visitors, only 35% used the alarm clock. We can calculate the z-score and use it to find the probability.

First, we calculate the standard deviation of the sample proportion using the formula:

Standard deviation = sqrt((p * (1 - p)) / n)

where p is the population proportion and n is the sample size. In this case, p is 0.38 and n is 700.

Once we have the standard deviation, we can calculate the z-score using the formula:

Z-score = (sample proportion - population proportion) / standard deviation

For this problem, the sample proportion is 0.35, the population proportion is 0.38, and the standard deviation is calculated in the previous step. Once we have the z-score, we can use a standard normal distribution table or a calculator to find the probability that the sample proportion is less than 0.35.