When Ella commutes to work, the amount of time it takes her to arrive is normally distributed with a mean of 34 minutes and a standard deviation of 1.5 minutes. Out of the 210 days that Hia commutes to work per year, how many times would her commute be between 28 and is minutes, to the nearest whole number?

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When Ella commutes to work, the amount of time it takes her to arrive is normally distributed with a mean of 34 minutes and a standard deviation of 1.5 minutes. Out of the 210 days that Hia commutes to work per year, how many times would her commute be between 28 and is minutes, to the nearest whole number?

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## Step 1: First, we need to convert the commute times of 28 and 38 minutes into z-scores. The z-score is a measure of how many standard deviations an element is from the mean. The formula to calculate the z-score is: ### \(Z = \frac{X - \mu}{\sigma}\) where \(X\) is the value we are interested in, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. ## Step 2: We calculate the z-scores for 28 and 38 minutes. For 28 minutes: ### \(Z_{28} = \frac{28 - 34}{4.5} = -1.33\) For 38 minutes: ### \(Z_{38} = \frac{38 - 34}{4.5} = 0.89\) ## Step 3: Next, we need to find the probability that the commute time is between these two z-scores. We can use a standard normal distribution table or a calculator to find these probabilities. The probability for \(Z_{28}\) is 0.0925 and for \(Z_{38}\) is 0.8133. The probability that the commute time is between 28 and 38 minutes is the difference between these two probabilities. ### \(P = P(Z_{38}) - P(Z_{28}) = 0.8133 - 0.0925 = 0.7208\) ## Step 4: Finally, we multiply this probability by the total number of days Fila commutes to work to find the number of days her commute would be between 28 and 38 minutes. ### \(N = P \times \text{Total days} = 0.7208 \times 210 = 151.37\) Since we cannot have a fraction of a day, we round this to the nearest whole number.

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