Given that z is a standard normal random variable, compute the following probabilities (to 4 decimals).
P(z -1.0)
P(z -1.0)
P(z -1.5)
P(z -2.5)
P(-3 < z 0)
P(z <-1.0) = 0.1587
P(z <-1.5) = 0.0668
P(z <-2.5) = 0.0062
P(-3 < z < 0) = 0.500-0.0013
Jim Tree sells Christmas trees. The mean length of the trees purchased was 68 inches with a standard deviation of 10 inches. Jim wants to know what per cent of his sales were more than 84 inches tall. He can use the standard normal distribution to help him.
To compute the probabilities, we will use the standard normal distribution table (also known as the Z-table).
1. P(z < -1.0):
From the Z-table, we find the probability associated with a z-score of -1.0 is 0.1587.
2. P(z < -1.5):
From the Z-table, we find the probability associated with a z-score of -1.5 is 0.0668.
3. P(z < -2.5):
From the Z-table, we find the probability associated with a z-score of -2.5 is 0.0062.
4. P(-3 < z < 0):
To find the probability of a z-score between -3 and 0, we subtract the probability of z < 0 from the probability of z < -3.
P(z < 0) = 0.5 (since the standard normal distribution is symmetric around 0).
From the Z-table, we find the probability associated with a z-score of -3 is 0.0013.
Therefore, P(-3 < z < 0) = P(z < 0) - P(z < -3) = 0.5 - 0.0013 = 0.4987 (rounded to 4 decimal places).
So, the probabilities are as follows:
P(z < -1.0) = 0.1587
P(z < -1.5) = 0.0668
P(z < -2.5) = 0.0062
P(-3 < z < 0) = 0.4987
To compute these probabilities, we need to use the standard normal distribution table or a statistical calculator.
1. P(z < -1.0):
To find this probability, we look up the value -1.0 in the standard normal distribution table. The table provides the area under the standard normal curve to the left of the given value. In this case, the table value is 0.1587 (rounded to four decimals). Therefore, P(z < -1.0) = 0.1587.
2. P(z < -1.5):
Similar to the previous case, we look up the value -1.5 in the standard normal distribution table. The entry in the table is 0.0668. Therefore, P(z < -1.5) = 0.0668.
3. P(z < -2.5):
Again, we look up the value -2.5 in the standard normal distribution table. The table value is 0.0062. So, P(z < -2.5) = 0.0062.
4. P(-3 < z < 0):
To find this probability, we need to find the area under the standard normal curve between -3 and 0. We first find the probability of z being less than 0 using the standard normal distribution table. The table entry for 0 is 0.5000. Next, we find the probability of z being less than -3, which is 0.0013 from the table. To find the probability between -3 and 0, we subtract the probability of z being less than -3 from the probability of z being less than 0: P(-3 < z < 0) = 0.5000 - 0.0013 = 0.4987.
So, the probabilities are:
P(z < -1.0) = 0.1587
P(z < -1.5) = 0.0668
P(z < -2.5) = 0.0062
P(-3 < z < 0) = 0.4987