Question

An article claimed that only 39% 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 500 and checked whether they used their room alarm clock. Of the sampled visitors, only 35% used the alarm clock. What is the approximate probability that less than 35% of the sample used the alarm clock (rounded to the nearest hundreth)?

a. 0.11
b. $0.05
c. 0.03
d. $0.08

Answer 1

To find the probability that less than 35% of the sample used the alarm clock, use the normal distribution approximation.

Step-by-step explanation:

To find the probability that less than 35% of the sample used the alarm clock, we can use the normal distribution approximation. First, we need to calculate the standard deviation of the sample proportion. The formula for the standard deviation is sqrt((p*(1-p))/n), where p is the hypothesized proportion (0.39) and n is the sample size (500). Plugging in the values, we get sqrt((0.39*(1-0.39))/500) = 0.025.

Next, we calculate the z-score for the observed proportion. The z-score formula is (observed proportion - hypothesized proportion) / standard deviation. So, (0.35 - 0.39) / 0.025 = -1.6.

Using a standard normal distribution table or calculator, we can find the probability that a z-score is less than -1.6, which is approximately 0.0559. Therefore, the approximate probability that less than 35% of the sample used the alarm clock is 0.06 (rounded to the nearest hundredth), so the answer is 0.06.