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 :
We are asked to find the probability that a single randomly selected value from a population that follows a normal distribution with a mean ((\mu)) of 15.7 and a standard deviation ((\sigma)) of 1.5 is less than 16.7. This is essentially asking us to find the probability (P(x < 16.7)), where (x = 16.7).
Step 1: Calculate the Z-score
To standardize the value of 16.7 in terms of the Z-score, we can use the formula:
[
z = \frac{x – \mu}{\sigma}
]
Where:
- (x = 16.7) is the value we are interested in,
- (\mu = 15.7) is the population mean, and
- (\sigma = 1.5) is the population standard deviation.
Plugging in the values:
[
z = \frac{16.7 – 15.7}{1.5} = \frac{1.0}{1.5} = 0.6667
]
Thus, the Z-score is approximately 0.67 when rounded to two decimal places.
Step 2: Find the Probability
Once we have the Z-score, we can find the probability that a randomly selected value from this normal distribution is less than 16.7 by using the standard normal distribution table (Z-table) or a calculator.
The Z-table provides the cumulative probability for a given Z-score. For (z = 0.67), the cumulative probability (i.e., the area under the standard normal curve to the left of (z = 0.67)) is approximately 0.7486.
Conclusion
Thus, the probability that a randomly selected value is less than 16.7 is:
[
P(x < 16.7) \approx 0.7486
]
Explanation
In a normal distribution, the Z-score represents how many standard deviations a value is away from the mean. By transforming the original value into a Z-score, we can use the standard normal distribution, which has a mean of 0 and a standard deviation of 1, to calculate the probability. The Z-score of 0.67 tells us that 16.7 is 0.67 standard deviations above the mean of 15.7. The Z-table then provides the cumulative probability to the left of this Z-score, which corresponds to the probability of selecting a value less than 16.7 from the population.
In summary, the probability is approximately 0.7486 or 74.86%. This means there is a 74.86% chance that a randomly selected value from this population will be less than 16.7.
