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At a local restaurant, the amount of time that customers have to wait for their food is normally distributed with a mean of 22 minutes and a standard deviation of 3 minutes. If you visit that restaurant 37 times this year, what is the expected number of times that you would expect to wait between 19 minutes and 23 minutes, to the nearest whole number?

At a local restaurant, the amount of time that customers have to wait for their food is normally distributed with a mean of 2222 minutes and a standard deviation of 33 minutes. If you visit that restaurant 3737 times this year, what is the expected number of times that you would expect to wait between 1919 minutes and 2323 minutes, to the nearest whole number?

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Q. At a local restaurant, the amount of time that customers have to wait for their food is normally distributed with a mean of 2222 minutes and a standard deviation of 33 minutes. If you visit that restaurant 3737 times this year, what is the expected number of times that you would expect to wait between 1919 minutes and 2323 minutes, to the nearest whole number?
  1. Given Information: We are given that the wait time for food at a local restaurant is normally distributed with a mean μ\mu of 2222 minutes and a standard deviation σ\sigma of 33 minutes. We want to find the probability of waiting between 1919 and 2323 minutes.
  2. Standardize Wait Times: First, we need to standardize the wait times of 1919 minutes and 2323 minutes using the z-score formula: z=(X−μ)/σz = (X - \mu) / \sigma, where XX is the value from the normal distribution, μ\mu is the mean, and σ\sigma is the standard deviation.
  3. Calculate z-score for 1919 minutes: Calculate the z-score for 1919 minutes: z=(19−22)/3=−3/3=−1z = (19 - 22) / 3 = -3 / 3 = -1.
  4. Calculate z-score for 2323 minutes: Calculate the z-score for 2323 minutes: z=23−223=13≈0.33z = \frac{23 - 22}{3} = \frac{1}{3} \approx 0.33.
  5. Lookup z-scores in table: Now, we look up these zz-scores in the standard normal distribution table or use a calculator to find the probabilities. The probability of zz being less than −1-1 is approximately 0.15870.1587, and the probability of zz being less than 0.330.33 is approximately 0.62930.6293.
  6. Find probability of waiting: To find the probability of waiting between 1919 and 2323 minutes, we subtract the probability of waiting less than 1919 minutes from the probability of waiting less than 2323 minutes: P(19 < X < 23) = P(X < 23) - P(X < 19) = 0.6293 - 0.1587 = 0.4706.
  7. Calculate expected number of times: Now, we multiply this probability by the number of visits to find the expected number of times you would wait between 1919 and 2323 minutes: Expected number of times = Total visits ×\times Probability = 37×0.470637 \times 0.4706.
  8. Round to nearest whole number: Calculate the expected number of times: Expected number of times = 37×0.4706≈17.412237 \times 0.4706 \approx 17.4122.
  9. Round to nearest whole number: Calculate the expected number of times: Expected number of times = 37×0.4706≈17.412237 \times 0.4706 \approx 17.4122.Since we cannot visit a fraction of a time, we round to the nearest whole number: Expected number of times ≈17\approx 17 (to the nearest whole number).

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