The lifetime of a SuperTough AAA battery is normally distributed with mean of 28.5 hours and standard deviation of 5.3 hours. For a battery selected at random, what is the probability that the lifetime will be 34 hours or less?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
To find the probability that the lifetime of a AAA battery will be 34 hours or less, we can use the concept of standardizing a random variable and applying the cumulative distribution function (CDF).
The CDF gives us the probability that a random variable is less than or equal to a certain value. In this case, we want to find the probability that the lifetime of a AAA battery is 34 hours or less.
Step 1: Standardize the value
To standardize the value, we need to calculate the z-score using the formula:
z = (x - μ) / σ
where
x = the value we want to standardize (34 hours in this case)
μ = mean of the distribution (28.5 hours)
σ = standard deviation of the distribution (5.3 hours)
z = (34 - 28.5) / 5.3
Step 2: Look up the probability
After calculating the z-score, we can now look up the probability using a standard normal distribution table or a calculator.
P(Z ≤ z) gives the probability that a standard normal random variable is less than or equal to the z-score.
In this case, we want to find P(Z ≤ z), where z is the calculated z-score.
Step 3: Interpret the result
The probability obtained from the table or calculator represents the probability that the lifetime of a SuperTough AAA battery is 34 hours or less.
Please note that in this case, we assume that the battery lifetimes are normally distributed.