A psychologist studied the time x (in seconds) it took a subject to complete a pencil and paper maze while in the presence of a floral scent. Suppose x has a normal distribution with mean 70 seconds and standard deviation 15 seconds. Find P(X<58)
Z = (score-mean)/SD = (58-70)/15 = ?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
To find P(X < 58), we need to convert the values to a standard normal distribution using z-scores and then look up the corresponding probability in a standard normal distribution table.
Step 1: Calculate the z-score
The z-score formula is:
z = (x - μ) / σ
Where:
x = the value we want to find the probability for (58 in this case)
μ = the mean of the distribution (70 seconds)
σ = the standard deviation of the distribution (15 seconds)
Substituting the values:
z = (58 - 70) / 15
z = -12 / 15
z = -0.8
Step 2: Look up the probability
Now we need to find the probability associated with the z-score of -0.8. We can use a standard normal distribution table or a calculator to find this.
Using a standard normal distribution table, we can find the cumulative probability associated with a z-score of -0.8. The cumulative probability is the probability of getting a value less than the given z-score.
Looking up the value in the table, we find that the cumulative probability corresponding to -0.8 is approximately 0.2119.
Step 3: Interpret the result
The probability of X being less than 58 seconds, P(X < 58), is approximately 0.2119 or 21.19%.
Therefore, there is a 21.19% chance that a subject will complete the maze in less than 58 seconds in the presence of a floral scent, according to the given normal distribution with mean 70 seconds and standard deviation 15 seconds.