Answer :
Final answer:
Approximately 9.85% of New York City commutes are less than 29 minutes, 23.8% are between 29 and 35 minutes, and 62.56% are between 29 and 42 minutes.
Explanation:
a. To find the percent of New York City commutes that are less than 29 minutes, we need to find the probability that a commute is less than 29 minutes using the normal distribution. We can use the z-score formula: z = (x - μ) / σ, where x is the given value, μ is the mean, and σ is the standard deviation. Substituting the values, we get z = (29 - 37.9) / 6.9 = -1.29. Looking up the z-value in the z-table, we find the corresponding cumulative probability to be 0.0985, or 9.85%. Therefore, approximately 9.85% of the New York City commutes are for less than 29 minutes.
b. To find the percent of commutes between 29 and 35 minutes, we need to find the difference in the probabilities for both values. Using the z-score formula again for each value, we get z1 = (29 - 37.9) / 6.9 = -1.29 and z2 = (35 - 37.9) / 6.9 = -0.42. Using the z-table, we find the cumulative probabilities for both z-values: P(z1) ≈ 0.0985 and P(z2) ≈ 0.3365. The difference between these probabilities is approximately 0.238, or 23.8%. Therefore, approximately 23.8% of the New York City commutes are between 29 and 35 minutes.
c. To find the percent of commutes between 29 and 42 minutes, we can use a similar approach as in part b. Find the z-values for both values: z1 = (29 - 37.9) / 6.9 = -1.29 and z2 = (42 - 37.9) / 6.9 = 0.59. Using the z-table, we find the cumulative probabilities for both z-values: P(z1) ≈ 0.0985 and P(z2) ≈ 0.7241. The difference between these probabilities is approximately 0.6256, or 62.56%. Therefore, approximately 62.56% of the New York City commutes are between 29 and 42 minutes.
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