Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion in the smaller area (.23) and its Z score. Use the equation below.
Z = (score-mean)/SD
Suppose a rod is chosen at random from all the rods produced by the company. There is a 23% probability that the rod is longer than:
Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.
Z = (score-mean)/SD
The z-score can be calculated using the following formula:
z = (x - μ) / σ
where:
z is the z-score
x is the value we want to find the probability for (in this case, the length of the rod)
μ is the mean of the distribution (121.2 cm)
σ is the standard deviation of the distribution (1.8 cm)
Step 1: Convert the probability to a z-score.
To find the z-score corresponding to a 23% probability, we need to find the z-score that has an area of 1 - 0.23 = 0.77 to its left. This corresponds to the z-score that marks the 77th percentile.
Step 2: Look up the z-score in a standard normal distribution table or use a calculator.
Using a standard normal distribution table or a calculator, we find that the z-score for the 77th percentile is approximately 0.60 (rounded to 2 decimal places).
Step 3: Solve for x (the length of the rod).
Now that we have the z-score, we can rearrange the formula to solve for x:
x = z * σ + μ
x = 0.60 * 1.8 + 121.2
x ≈ 122.08
Therefore, the length of the rod that has a 23% probability of being longer is approximately 122.1 cm (rounded to 1 decimal place).