Answer :
To solve this problem, we need to find the area under the standard normal distribution curve that is to the left of [tex]\( z = -0.64 \)[/tex] and to the right of [tex]\( z = 1.64 \)[/tex].
Here's how we can determine these areas:
1. Find the area to the left of [tex]\( z = -0.64 \)[/tex]:
- The cumulative distribution function (CDF) of a standard normal distribution gives the probability that a standard normal random variable is less than a given value. For [tex]\( z = -0.64 \)[/tex], the CDF tells us the probability that [tex]\( z \)[/tex] is less than [tex]\(-0.64\)[/tex]. This represents the area under the curve to the left of [tex]\( z = -0.64 \)[/tex].
2. Find the area to the right of [tex]\( z = 1.64 \)[/tex]:
- The area to the right of a z-score can be found by taking `1 - CDF(z)`, where CDF(z) gives the area to the left. For [tex]\( z = 1.64 \)[/tex], the area to the right is calculated as the complement of the area to the left, which is [tex]\( 1 - \text{CDF}(1.64) \)[/tex].
3. Add both areas:
- The total area under the curve to the specified regions (to the left of [tex]\( z = -0.64\)[/tex] and to the right of [tex]\( z = 1.64\)[/tex]) is the sum of the two areas found above.
By carrying out these calculations, we find the total area to be [tex]\( 0.3116 \)[/tex].
So, the area under the standard normal distribution curve to the left of [tex]\( z = -0.64 \)[/tex] and to the right of [tex]\( z = 1.64 \)[/tex] is 0.3116.
Here's how we can determine these areas:
1. Find the area to the left of [tex]\( z = -0.64 \)[/tex]:
- The cumulative distribution function (CDF) of a standard normal distribution gives the probability that a standard normal random variable is less than a given value. For [tex]\( z = -0.64 \)[/tex], the CDF tells us the probability that [tex]\( z \)[/tex] is less than [tex]\(-0.64\)[/tex]. This represents the area under the curve to the left of [tex]\( z = -0.64 \)[/tex].
2. Find the area to the right of [tex]\( z = 1.64 \)[/tex]:
- The area to the right of a z-score can be found by taking `1 - CDF(z)`, where CDF(z) gives the area to the left. For [tex]\( z = 1.64 \)[/tex], the area to the right is calculated as the complement of the area to the left, which is [tex]\( 1 - \text{CDF}(1.64) \)[/tex].
3. Add both areas:
- The total area under the curve to the specified regions (to the left of [tex]\( z = -0.64\)[/tex] and to the right of [tex]\( z = 1.64\)[/tex]) is the sum of the two areas found above.
By carrying out these calculations, we find the total area to be [tex]\( 0.3116 \)[/tex].
So, the area under the standard normal distribution curve to the left of [tex]\( z = -0.64 \)[/tex] and to the right of [tex]\( z = 1.64 \)[/tex] is 0.3116.
