Answer:
Explanation:
To answer these questions, we need to use the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. We can convert the original distribution to the standard normal distribution using the formula:
z = (x - μ) / σ
where z is the standard normal score, x is the original value, μ is the mean, and σ is the standard deviation.
To find the probability that a given 1-ml sample will contain at most 100 bacteria, we need to find the area under the normal curve to the left of 100. We can calculate the z-score for 100 as:
z = (100 - 85) / 9 = 1.67
Using a standard normal table or a calculator, we can find that the area to the left of 1.67 is 0.9525. Therefore, the probability of a 1-ml sample containing at most 100 bacteria is 0.9525.
To find the probability that a given 1-ml sample will contain at least 75 bacteria, we need to find the area under the normal curve to the right of 75. We can calculate the z-score for 75 as:
z = (75 - 85) / 9 = -1.11
Using a standard normal table or a calculator, we can find that the area to the right of -1.11 (which is the same as the area to the left of 1.11) is 0.8665. Therefore, the probability of a 1-ml sample containing at least 75 bacteria is 0.8665.
To find the probability that a given 1-ml sample will contain between 90 and 95 bacteria, we need to find the area under the normal curve between the z-scores for 90 and 95. We can calculate the z-scores as:
z1 = (90 - 85) / 9 = 0.56
z2 = (95 - 85) / 9 = 1.11
Using a standard normal table or a calculator, we can find that the area to the left of 0.56 is 0.7123 and the area to the left of 1.11 is 0.8665. Therefore, the area between 0.56 and 1.11 is:
0.8665 - 0.7123 = 0.1542
Therefore, the probability of a 1-ml sample containing between 90 and 95 bacteria is 0.1542.
To find the probability of a 1-ml sample containing below 70 or above 100 bacteria, we need to find the area under the normal curve to the left of 70 and the area to the right of 100, and add them together. We can calculate the z-scores for 70 and 100 as:
z1 = (70 - 85) / 9 = -1.67
z2 = (100 - 85) / 9 = 1.67
Using a standard normal table or a calculator, we can find that the area to the left of -1.67 is 0.0475 and the area to the right of 1.67 is 0.0475. Therefore, the probability of a 1-ml sample containing below 70 or above 100 bacteria is:
0.0475 + 0.0475 = 0.095
Therefore, the probability of a 1-ml sample containing below 70 or above 100 bacteria is 0.095.