Sure, let's determine the probability [tex]\( P(-2.87 < z < 0.38) \)[/tex] for a standard normal distribution step-by-step.
1. Identify the z-scores:
- The lower z-score is [tex]\( z = -2.87 \)[/tex].
- The upper z-score is [tex]\( z = 0.38 \)[/tex].
2. Determine the cumulative distribution function (CDF) values for the z-scores:
- The CDF gives us the probability that a standard normal random variable is less than or equal to a particular value.
- For [tex]\( z = -2.87 \)[/tex]:
[tex]\[
\Phi(-2.87) \approx 0.002052
\][/tex]
- For [tex]\( z = 0.38 \)[/tex]:
[tex]\[
\Phi(0.38) \approx 0.648027
\][/tex]
3. Calculate the probability that [tex]\( z \)[/tex] is between -2.87 and 0.38:
- This probability is found by subtracting the CDF value at [tex]\( z = -2.87 \)[/tex] from the CDF value at [tex]\( z = 0.38 \)[/tex]:
[tex]\[
P(-2.87 < z < 0.38) = \Phi(0.38) - \Phi(-2.87)
\][/tex]
- Substituting the values we have:
[tex]\[
P(-2.87 < z < 0.38) = 0.648027 - 0.002052
\][/tex]
[tex]\[
P(-2.87 < z < 0.38) \approx 0.645975
\][/tex]
Therefore, the probability [tex]\( P(-2.87 < z < 0.38) \)[/tex] in a standard normal distribution is approximately 0.645975.
