Answer:
o solve these problems, we will use the standard normal distribution table.
a. To find the probability that someone consumed more than 41 gallons of bottled water, we need to find the area under the curve to the right of 41. To do this, we convert 41 into a z-score using the formula:
z = (x - µ) / σ
Where x is the value we want to convert to a z-score (41), µ is the mean (31.2), and σ is the standard deviation (10).
z = (41 - 31.2) / 10 = 0.98
Using the standard normal distribution table, we can find the probability associated with a z-score of 0.98. The probability is 0.8365.
Therefore, the probability that someone consumed more than 41 gallons of bottled water is 0.8365.
b. To find the probability that someone consumed between 20 and 30 gallons of bottled water, we need to find the area under the curve between 20 and 30. First, we convert the values into z-scores.
For 20 gallons:
z1 = (20 - 31.2) / 10 = -1.12
For 30 gallons:
z2 = (30 - 31.2) / 10 = -0.12
Using the standard normal distribution table, we can find the probabilities associated with these z-scores.
The probability associated with z = -0.12 is 0.4535.
The probability associated with z = -1.12 is 0.1314.
To find the probability between these two z-scores, we subtract the smaller probability from the larger one.
0.4535 - 0.1314 = 0.3221
Therefore, the probability that someone consumed between 20 and 30 gallons of bottled water is 0.3221.
c. To find the probability that someone consumed less than 20 gallons of bottled water, we need to find the area under the curve to the left of 20. We convert 20 into a z-score.
z = (20 - 31.2) / 10 = -1.12
Using the standard normal distribution table, we can find the probability associated with a z-score of -1.12. The probability is 0.1314.
Therefore, the probability that someone consumed less than 20 gallons of bottled water is 0.1314.
d. To find the value for which 90% of people consumed less than, we need to find the z-score associated with a cumulative probability of 0.90.
Using the standard normal distribution table, we look for the value closest to 0.90. The closest value is 0.8997, which is associated with a z-score of 1.28.
To find the value of x, we rearrange the z-score formula:
x = z * σ + µ
Substituting the values we have:
x = 1.28 * 10 + 31.2 = 44.8
Therefore, 90% of people consumed less than 44.8 gallons of bottled water.
Explanation: