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 away from the mean. It is calculated using the formula:
[ Z = \frac{X – \mu}{\sigma} ]
Given:
- ( X = 16.7 )
- ( \mu = 15.7 )
- ( \sigma = 1.5 )
Substitute these values into the formula:
[ Z = \frac{16.7 – 15.7}{1.5} = \frac{1}{1.5} \approx 0.6667 ]
Rounded to two decimal places, the Z-score is:
[ Z \approx 0.67 ]
2. Determine the probability corresponding to the Z-score:
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 associated with a Z-score of 0.67.
Using a standard normal distribution table or a Z-score calculator, we find that the cumulative probability for a Z-score of 0.67 is approximately 0.7486.
3. Interpret the result:
A cumulative probability of 0.7486 means that there is a 74.86% chance that a randomly selected value from this population will be less than 16.7.
Explanation:
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric around the mean. The Z-score standardizes a value, allowing us to determine its position relative to the mean in terms of standard deviations. By converting the raw score to a Z-score and consulting the standard normal distribution, we can find the probability of observing a value less than or equal to the given score.
The shaded area under the curve to the left of Z = 0.67 represents the cumulative probability of approximately 74.86%.
Conclusion:
By calculating the Z-score for the value 16.7 and determining the corresponding cumulative probability, we find that there is a 74.86% probability that a single randomly selected value from this normal distribution will be less than 16.7.