Final answer:
To find the probability that the temperature of the coffee is between 150 degrees and 154 degrees, we calculate the z-scores for both values and use the standard normal distribution table. The probability is 0.2881, or 28.81%. To find the probability that the temperature is more than 164 degrees, we calculate the z-score for 164 degrees and use the standard normal distribution table. The probability is 0.0026, or 0.26%.
Step-by-step explanation:
To solve this problem, we'll use the z-score formula and the standard normal distribution table. The z-score formula is z = (x - μ) / σ. where x is the value we're interested in, μ is the mean, and σ is the standard deviation. a. To find the probability that the temperature is between 150 degrees and 154 degrees, we need to calculate the z-scores for both values and then find the area between them. The z-score for 150 degrees is z = (150 - 150) / 5 = 0. The z-score for 154 degrees is z = (154 - 150) / 5 = 0.8. Using the standard normal distribution table, we can find that the area to the left of 0 is 0.5, and the area to the left of 0.8 is 0.7881. So the area between them is 0.7881 - 0.5 = 0.2881. Therefore, the probability that the temperature of the coffee is between 150 degrees and 154 degrees is 0.2881, or 28.81%. b. To find the probability that the temperature is more than 164 degrees, we need to calculate the z-score for 164 degrees and then find the area to the right of that z-score. The z-score for 164 degrees is z = (164 - 150) / 5 = 2.8. Using the standard normal distribution table, we can find that the area to the left of 2.8 is 0.9974. So the area to the right of 2.8 is 1 - 0.9974 = 0.0026. Therefore, the probability that the coffee temperature is more than 164 degrees is 0.0026, or 0.26%.