Answer :
To solve this problem, we need to determine the value of [tex]z[/tex] for which the standard normal distribution has an area of 0.0582 to the right of [tex]z[/tex]. This requires us to look up a standard normal distribution table or use a statistical calculator that provides the [tex]z[/tex]-score for a given cumulative probability.
Understand the Standard Normal Distribution: The standard normal distribution is a continuous probability distribution that is symmetrical around the mean, which is 0, with a standard deviation of 1. It's frequently used in statistics to model real-world data.
Focus on the Area to the Right: The problem specifies the area to the right of [tex]z[/tex] is 0.0582. This means we are looking for a [tex]z[/tex]-score such that the probability that a standard normal variable is greater than [tex]z[/tex] is 0.0582.
Finding the [tex]z[/tex]-score: To find this, you can either use a z-score table or an online calculator which provides inverse cumulative probabilities.
- The cumulative probability to the left of this [tex]z[/tex]-score would be [tex]1 - 0.0582 = 0.9418[/tex].
- When you look up this cumulative probability in a standard normal distribution table or use a calculator, you find that a [tex]z[/tex]-score of approximately [tex]1.57[/tex] corresponds to a cumulative probability of 0.9418.
Thus, the correct answer is option c. 1.57.
In summary, for a standard normal distribution, if the area to the right of [tex]z[/tex] is 0.0582, then the [tex]z[/tex]-score is approximately 1.57.
