Certainly! Let's delve through the steps to find the score [tex]\( a \)[/tex] given the probability [tex]\( P(Z < a) \)[/tex].
1. Understanding the Problem:
- We know from the problem statement that [tex]\( P(Z < a) = 0.042716 \)[/tex].
- Our goal is to find the value of [tex]\( a \)[/tex] such that the probability of the standard normal variable [tex]\( Z \)[/tex] being less than [tex]\( a \)[/tex] is 0.042716.
2. Cumulative Distribution Function (CDF):
- The cumulative distribution function (CDF) of a standard normal distribution tells us the probability that the variable takes a value less than or equal to [tex]\( a \)[/tex].
- Mathematically, for the standard normal distribution [tex]\( Z \)[/tex], the CDF is denoted as [tex]\( \Phi(a) \)[/tex], where [tex]\( \Phi(a) \)[/tex] gives [tex]\( P(Z < a) \)[/tex].
3. Inverse CDF (Percent Point Function, PPF):
- To find [tex]\( a \)[/tex] for a given probability [tex]\( P(Z < a) \)[/tex], we need to use the inverse cumulative distribution function (also known as the percent point function, PPF).
- This function, typically denoted as [tex]\( \Phi^{-1}(p) \)[/tex], finds the [tex]\( a \)[/tex] such that [tex]\( P(Z < a) = p \)[/tex].
4. Given Probability:
- In this case, the probability is [tex]\( p = 0.042716 \)[/tex].
5. Finding the Score [tex]\( a \)[/tex]:
- Utilizing the inverse cumulative distribution function, [tex]\( a \)[/tex] is the score where [tex]\( \Phi(a) = 0.042716 \)[/tex].
- In statistical software or standard normal distribution tables, this is commonly denoted as [tex]\( \mathrm{PPF}(0.042716) \)[/tex].
6. Result:
- Through calculation using the inverse cumulative distribution function, the resulting value for [tex]\( a \)[/tex] is approximately [tex]\( -1.7200024293005731 \)[/tex].
### Final Answer:
The score [tex]\( a \)[/tex] such that [tex]\( P(Z < a) = 0.042716 \)[/tex] is approximately [tex]\( -1.7200024293005731 \)[/tex].
### Summary:
By using the inverse cumulative distribution function for the given probability, we were able to find the corresponding score [tex]\( a \)[/tex]. This reverse lookup from probability to score is essential in statistical analysis and helps us understand the corresponding quantiles in standard normal distribution.
