A manufacturer knows that their items have a normally distributed length, with a mean of 7.3 inches, and standard deviation of 0.6 inches.
If one item is chosen at random, what is the probability that it is less than 5.5 inches long?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/percentage related to the Z score.
-6.66
To find the probability that a randomly chosen item is less than 5.5 inches long, we need to calculate the z-score and then use the standard normal distribution table.
The z-score is calculated as:
z = (x - μ) / σ
where x is the value we are interested in (in this case, 5.5 inches), μ is the mean (7.3 inches), and σ is the standard deviation (0.6 inches).
Plugging in the values, we get:
z = (5.5 - 7.3) / 0.6
z = -3
We can now look up the z-score -3 in the standard normal distribution table. However, since the z-score of -3 is beyond the range of most standard normal distribution tables, we can make an assumption based on the symmetry of the normal distribution:
P(Z < -3) ≈ P(Z > 3)
Using a standard normal distribution table, we find that P(Z > 3) is approximately 0.0013.
Therefore, the probability that a randomly chosen item is less than 5.5 inches long is approximately 0.0013.
To find the probability that a randomly chosen item is less than 5.5 inches long, we need to calculate the z-score and then use a standard normal distribution table (also known as a z-table) or a statistical calculator.
Step 1: Calculate the z-score.
The z-score measures the number of standard deviations a data point is away from the mean. In this case, we want to find the z-score for 5.5 inches using the given mean and standard deviation.
The formula for calculating the z-score is:
z = (x - μ) / σ
Where:
x = the given value (5.5 inches)
μ = the mean length (7.3 inches)
σ = the standard deviation (0.6 inches)
Substituting the values into the formula, we get:
z = (5.5 - 7.3) / 0.6 = -3
Step 2: Look up the z-score.
Next, we need to find the probability associated with this z-score. We can use a standard normal distribution table or a statistical calculator to obtain this probability.
For a z-score of -3, we can use the z-table to find the area (probability) to the left of -3. The table provides the area under the standard normal distribution curve up to a certain z-score. In this case, a z-score of -3 indicates that we need to look up the probability for the area to the left of -3.
Using the z-table or a calculator, we find that the probability (area) to the left of -3 is approximately 0.0013.
Step 3: Calculate the final probability.
Since we want the probability that the item is less than 5.5 inches, and the standard normal distribution table provides the probability to the left of a given z-score, we can use the following formula to calculate the final probability:
P(X < 5.5) = P(Z < -3)
P(Z < -3) = 0.0013 (from Step 2)
Therefore, the probability that a randomly chosen item is less than 5.5 inches long is approximately 0.0013, or 0.13%.