Final answer:
To find the probability that a normally distributed variable x is between 134.4 and 140.1, convert the x-values to z-scores, find the cumulative distribution function (CDF) values for these z-scores, and subtract them to get the probability.
Step-by-step explanation:
The student asked for help with finding the probability that a normally distributed variable x falls between two values, given the mean (μ) and the standard deviation (σ) of the distribution. Assuming that x follows a normal distribution with a mean of 137.0 and a standard deviation of 5.3, we need to find the probability that x lies between 134.4 and 140.1. This type of problem is commonly solved using standardization, where we convert the x-values into z-scores and then look up these scores in the standard normal distribution table or use a calculator for the cumulative distribution function (CDF).
First, we calculate the z-scores for 134.4 and 140.1:
For x=134.4:
z = (134.4 - 137.0) / 5.3
z ≈ -0.4906
For x=140.1:
z = (140.1 - 137.0) / 5.3
z ≈ 0.5849
Next, we use the standard normal distribution table or CDF calculator to find the area under the curve between these z-scores. Let's denote the CDF values for these z-scores as P(z ≈ -0.4906) and P(z ≈ 0.5849), respectively.
The desired probability is:
P(134.4 < x < 140.1) = P(z ≈ 0.5849) - P(z ≈ -0.4906)
The final step is to look up or calculate these probabilities and subtract them to get the probability area between 134.4 and 140.1, which we round to four decimal places as per the question's instructions.