The average number of calories in a 1.5 ounce chocolate bar is 225. Suppose that the distribution of calories is approximately normal with standard deviation =10. Find the probability that a randomly selected chocolate bar will have
a) Between 200 and 220 calories
To find the probability that a randomly selected chocolate bar will have between 200 and 220 calories, we first need to standardize the values using z-scores. The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
x = The value we want to standardize (in this case, 200 and 220)
μ = The mean of the distribution (average number of calories in a chocolate bar)
σ = The standard deviation of the distribution
In this case, the mean (μ) is 225 calories and the standard deviation (σ) is 10 calories.
Calculating the z-score for 200 calories:
z1 = (200 - 225) / 10 = -2.5
Calculating the z-score for 220 calories:
z2 = (220 - 225) / 10 = -0.5
Next, we need to use a standard normal distribution table or a calculator to find the probability corresponding to these z-scores. The probability that a randomly selected chocolate bar will have between 200 and 220 calories is equal to the difference between the two cumulative probabilities.
P(200 ≤ x ≤ 220) = P(z1 ≤ z ≤ z2)
Using the standard normal distribution table or calculator, we can find the cumulative probabilities corresponding to the z-scores -2.5 and -0.5. Subtracting the smaller probability from the larger will give us the final answer.