Consider the standard normal distribution with mean μ = 0, and standard
deviation σ = 1
a. What is the probability that an outcome z is greater than 2.20?
b. What is the probability that z is less than 1.1?
c. What is the probability that z is between -1.03 and 0.84?
d. What value of z cuts off the upper 23% of the standard normal distribution?
e. What value of z cuts off the lower
18% of the standard normal distribution?
Consult the Z (normal distribution) table in the back of your textbook.
a. Oh, so you want to know the probability that an outcome z is greater than 2.20? Well, that probability is so low it's practically non-existent! Like trying to find a unicorn in a haystack.
b. Now, the probability that z is less than 1.1? Oh, that's like finding money in your pants pocket that you forgot about. It's not impossible, but it's definitely a pleasant surprise.
c. Ah, the probability that z is between -1.03 and 0.84? Well, that's like trying to find some peace and quiet at a clown convention. It's possible, but it might be a bit chaotic.
d. The value of z that cuts off the upper 23% of the standard normal distribution? Ah, that's like finding the holy grail of normal distributions. It's like finding a clown with perfect timing - rare and elusive!
e. Lastly, the value of z that cuts off the lower 18% of the standard normal distribution? Well, that's like finding a clown who can juggle chainsaws - a bit dangerous and out of the ordinary, but definitely entertaining!
a. To find the probability that an outcome z is greater than 2.20, we need to calculate the area under the standard normal curve to the right of z = 2.20.
We can use a standard normal distribution table or a calculator to find this probability. From the standard normal distribution table, the area to the left of 2.20 is 0.9857. Since the total area under the curve is 1, the area to the right of 2.20 is 1 - 0.9857 = 0.0143.
Therefore, the probability that an outcome z is greater than 2.20 is approximately 0.0143.
b. To find the probability that z is less than 1.1, we need to calculate the area under the standard normal curve to the left of z = 1.1.
Using the standard normal distribution table, the area to the left of 1.1 is 0.8643.
Therefore, the probability that z is less than 1.1 is approximately 0.8643.
c. To find the probability that z is between -1.03 and 0.84, we need to calculate the area under the standard normal curve between these two values.
Using the standard normal distribution table, the area to the left of -1.03 is 0.1492, and the area to the left of 0.84 is 0.7995.
Therefore, the probability that z is between -1.03 and 0.84 is approximately 0.7995 - 0.1492 = 0.6503.
d. To find the value of z that cuts off the upper 23% of the standard normal distribution, we need to find the z-value corresponding to the area of 1-0.23=0.77.
Using the standard normal distribution table, we find that the z-value corresponding to an area of 0.77 is approximately 0.62.
Therefore, the value of z that cuts off the upper 23% of the standard normal distribution is approximately 0.62.
e. To find the value of z that cuts off the lower 18% of the standard normal distribution, we need to find the z-value corresponding to the area of 0.18.
Using the standard normal distribution table, we find that the z-value corresponding to an area of 0.18 is approximately -0.89.
Therefore, the value of z that cuts off the lower 18% of the standard normal distribution is approximately -0.89.
To solve these questions, we can use the standard normal distribution table, also known as the z-table. This table gives us the cumulative probability for different values of z.
a. To find the probability that an outcome z is greater than 2.20, we need to find the area under the curve to the right of 2.20. By looking up the corresponding value in the z-table, we find that the area to the left of 2.20 is 0.9857. Since the total area under the curve is 1, the area to the right of 2.20 is equal to 1 - 0.9857 = 0.0143, or 1.43%.
b. Similarly, to find the probability that z is less than 1.1, we look up the value 1.1 in the z-table and find the area to the left of it. The corresponding value is 0.8643. Therefore, the probability is 0.8643, or 86.43%.
c. To find the probability that z is between -1.03 and 0.84, we need to find the area between these two values. Using the z-table, we find that the area to the left of -1.03 is 0.1492, and the area to the left of 0.84 is 0.7995. Therefore, the area between -1.03 and 0.84 is 0.7995 - 0.1492 = 0.6503, or 65.03%.
d. To find the value of z that cuts off the upper 23% of the standard normal distribution, we need to find the z-value such that the area to its left is 1 - 0.23 = 0.77. Using the z-table, we look for the closest value to 0.77, which is 0.7704. The corresponding z-value is 0.71.
e. To find the value of z that cuts off the lower 18% of the standard normal distribution, we need to find the z-value such that the area to its left is 0.18. By looking up 0.18 in the z-table, we find that the corresponding z-value is -0.92.
In summary:
a. The probability that z is greater than 2.20 is 0.0143 or 1.43%.
b. The probability that z is less than 1.1 is 0.8643 or 86.43%.
c. The probability that z is between -1.03 and 0.84 is 0.6503 or 65.03%.
d. The z-value that cuts off the upper 23% is 0.71.
e. The z-value that cuts off the lower 18% is -0.92.