A population of values has a normal distribution with µ = 15.7 and s = 1.5. You intend to draw a random sample of size n = 18. First calculate z, round it to two (2) decimal places, then use the rounded z-score to determine the required probability accurate to four (4) decimal places. Find the probability that a single randomly selected value is less than 16.7. P(x < 16.7) = ?
The correct answer and explanation is :
To determine the probability that a single randomly selected value from a normally distributed population with a mean (µ) of 15.7 and a standard deviation (σ) of 1.5 is less than 16.7, we can follow these steps:
1. Calculate the z-score:
The z-score represents how many standard deviations a particular value (x) is from the mean and is calculated using the formula:
[ z = \frac{x – \mu}{\sigma} ]
Applying the given values:
[ z = \frac{16.7 – 15.7}{1.5} ]
This calculation yields a z-score of approximately 0.67.
2. Determine the cumulative probability:
The z-score of 0.67 indicates that the value 16.7 is 0.67 standard deviations above the mean. To find the probability that a randomly selected value is less than 16.7, we need to determine the cumulative probability corresponding to this z-score.
Using standard normal distribution tables or a calculator, a z-score of 0.67 corresponds to a cumulative probability of approximately 0.7486. This means there’s a 74.86% chance that a randomly selected value from this distribution is less than 16.7.
Explanation:
In a normal distribution, the mean divides the data into two equal halves. The standard deviation measures the average distance of data points from the mean. A z-score transforms a raw score into a standard score, indicating how many standard deviations the raw score is from the mean. By calculating the z-score for 16.7, we standardize it, allowing us to use standard normal distribution tables to find the corresponding cumulative probability.
Understanding z-scores and cumulative probabilities is crucial in statistics for assessing how typical or atypical a particular value is within a given distribution.