A large shipment of Granny Smith apples arrives at a grocery store. Weights of the apples are known to follow a normal distribution with mean 139 grams and standard deviation 9 grams.
1)If you select a random sample of 50 apples, what is the probability that the total weight of the apples is greater than 6853.9 grams?
6853.9/60 = 114.23 per apple, the score
Z = (score-mean)/SD
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 the probability that the total weight of the apples is greater than 6853.9 grams, we need to use a concept called the Central Limit Theorem.
The Central Limit Theorem states that when we take a random sample from a population with any distribution (including non-normal), as the sample size increases, the distribution of the sample means approaches a normal distribution.
In this case, since the weight of each apple is normally distributed with a mean of 139 grams and a standard deviation of 9 grams, the sum of the weights of a sample of 50 apples will also be normally distributed.
To find the probability, we need to standardize the variable X (the sum of the weights) and convert it into a standard normal distribution (Z) using the formula Z = (X - μ) / (σ / sqrt(n)), where X is the observed value, μ is the mean, σ is the standard deviation, and n is the sample size.
Using the given values:
μ = 139 grams (mean)
σ = 9 grams (standard deviation)
n = 50 (sample size)
X = 6853.9 grams (observed sum)
Z = (6853.9 - 139 * 50) / (9 * sqrt(50))
Calculating this equation, we find Z = 21.34.
Now, we need to find the probability that Z is greater than 21.34. We can use a standard normal distribution table or a statistical calculator to find this probability. Since the probability of a value being greater than a certain Z-value is the same as the probability of a value being less than the negative of that Z-value, we can find the probability of Z being less than -21.34 and then subtract it from 1.
Checking a standard normal distribution table or using a calculator, the probability of Z being less than -21.34 is practically zero (close to 0). Subtracting this probability from 1 gives us the answer.
Therefore, the probability that the total weight of the apples is greater than 6853.9 grams is practically 1 (100%).