The weights of individual nails produced by the 'Hammer 'n Nail' hardware company follow a normal distribution with the average nail weight being 5.1 grams and the standard deviation in nail weight being 1.6 grams. When a nail is produced it is discarded if it weighs more than 8.268 grams. Calculate the probability that a randomly selected nail is not discarded. Give your answer as a decimal to 4 decimal places.
Probability =
Z = (score-mean)/SD = (8.268-5.1)/1.6 = ?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to that Z score.
To calculate the probability that a randomly selected nail is not discarded, we need to find the area under the normal curve to the left of the threshold weight.
First, we need to standardize the threshold weight of 8.268 grams using the formula z = (x - μ) / σ, where x is the threshold weight, μ is the mean, and σ is the standard deviation.
z = (8.268 - 5.1) / 1.6
z = 1.93
Next, we need to find the cumulative probability to the left of this standardized value. We can use a standard normal distribution table or a calculator to find this value.
Using a standard normal distribution table or a calculator, we find that the cumulative probability to the left of z = 1.93 is approximately 0.9732.
Therefore, the probability that a randomly selected nail is not discarded is 1 - 0.9732 = 0.0268.
So, the probability is approximately 0.0268.