Final answer:
The probability that the Rest of the Night Mattress Store will sell between 53 and 67 mattresses on a given day is 83.8% when the daily demand is normally distributed with a mean of 60 and a standard deviation of 5.
Step-by-step explanation:
To calculate the probability that the Rest of the Night Mattress Store will sell between 53 and 67 mattresses on a given day, we can use the properties of normal distribution. Since the daily demand for mattresses is normally distributed with a mean (μ) of 60 and a standard deviation (σ) of 5, we can standardize the values of 53 and 67 to convert them into z-scores.
The z-score formula is given by: z = (X - μ) / σ, where X is the value we want to standardize. For X=53, the z-score is z = (53 - 60) / 5 = -1.4; and for X=67, it's z = (67 - 60) / 5 = 1.4.
After finding the z-scores, we can look up the values in a standard normal distribution table or use a calculator to find the probability corresponding to these z-scores. The probability of selling between 53 and 67 mattresses is the probability of being between the z-scores of -1.4 and 1.4.
Using a standard normal distribution table or calculator, we find that the probability of a z-score being less than -1.4 is approximately 0.0808, and the probability of a z-score being less than 1.4 is approximately 0.9192. To find the probability of being between the two z-scores, we subtract the smaller probability from the larger one:
Probability (53 ≤ X ≤ 67) = P(Z < 1.4) - P(Z < -1.4) = 0.9192 - 0.0808 = 0.8384
Thus, the probability of selling between 53 and 67 mattresses on a given day is 83.84%, or when rounded to the nearest tenth of a percent, it is 83.8%.