To find the value of [tex]\( c \)[/tex] such that [tex]\( P(Z < c) = 0.3748 \)[/tex] for a standard normal distribution, we need to determine the z-score corresponding to the cumulative probability of 0.3748. Here is the detailed step-by-step solution:
1. Understanding the Problem:
- We are given the cumulative probability [tex]\( P(Z < c) = 0.3748 \)[/tex].
- Our goal is to find the z-score [tex]\( c \)[/tex] such that the cumulative probability up to that z-score is 0.3748.
2. Standard Normal Distribution:
- The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
- The z-scores for the standard normal distribution are tabulated in z-tables, which display the cumulative probabilities.
3. Finding the Corresponding z-score:
- To find [tex]\( c \)[/tex], we need to locate the probability of 0.3748 in the z-table. Alternatively, modern computational methods are often used to find this value expediently, as looking up in tables can be imprecise and cumbersome.
4. Result Interpretation:
- The z-score that corresponds to a cumulative probability of 0.3748 is approximately [tex]\(-0.31916684109205823\)[/tex].
5. Rounding the Result:
- We are required to round this value to two decimal places.
- When rounding [tex]\(-0.31916684109205823\)[/tex] to two decimal places, we get [tex]\(-0.32\)[/tex].
Thus, the value of [tex]\( c \)[/tex] such that [tex]\( P(Z < c) = 0.3748 \)[/tex] is:
[tex]\[ c \approx -0.32 \][/tex]
So, [tex]\( c \)[/tex] rounded to two decimal places is [tex]\(-0.32\)[/tex].
