The heights of boys at a particular age follow a normal distribution with mean 150.3 and standard deviation 5cm. Find the probability that the height of a boy picked at random from the age group is i.more than 10 cm from the mean height.
95.4% of a normally distributed population lies within two s.d. of the mean
Z = (score-mean)/SD = (score-105.3)/5
Since you don't indicate the direction of deviation, the score could be 140.3 or 160.3. If you don't want to use the equation, the scores are ± 2Z.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability for the Z scores.
To find the probability that the height of a randomly picked boy from the age group is more than 10 cm from the mean height, we can apply the properties of the normal distribution.
Step 1: Calculate the standard deviation from the mean. Given that the standard deviation is 5 cm and the mean is 150.3 cm, we have σ = 5.
Step 2: Calculate the z-score for the given distance from the mean. The z-score formula is calculated using the formula z = (x - μ) / σ, where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
In this case, we want to find the probability for a distance of more than 10 cm from the mean. So, x = μ + 10. Plugging in the values, we have x = 150.3 + 10 = 160.3.
Now calculate the z-score: z = (160.3 - 150.3) / 5 = 2.
Step 3: Find the probability using the z-score. We can use a standard normal distribution table or a calculator to find the probability associated with the z-score.
Since we are looking for the probability of being more than 10 cm from the mean (to the right of the z-score on the normal distribution), we will subtract the cumulative probability from 0.5.
Using the standard normal distribution table, we can find the cumulative probability for z = 2, which is 0.9772.
Subtracting this value from 0.5 gives us the probability of being more than 10 cm from the mean: P(x > 160.3) = 0.5 - 0.9772 = -0.4772.
However, probabilities cannot be negative, so the correct probability is the absolute value of this negative value: P(x > 160.3) = 0.4772.
Therefore, the probability that the height of a randomly picked boy from the age group is more than 10 cm from the mean is approximately 0.4772.