Jackson is comparison shopping for orange juice. He created a table to help him decide which package was the best deal.Orange Ju
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Answer:
a) Mean 250 ml, standard deviation 1.2247 ml.
b) 74.14% probability that the volume of a single randomly chosen can differs from the target value by 1 ml or more.
c) 41.22% probability that the sample mean volume of a random sample of 6 cans differs from the target value by 1 ml or more
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this question:
a) Assume that the process mean is exactly equal to the target value, that is μ=250. What will be the mean and standard deviation of the sampling distribution of the sample mean x?
By the Central Limit Theorem, the mean is 250 ml and the standard deviation is ml.
b) What is the probability that the volume of a single randomly chosen can differs from the target value by 1 ml or more?
Greater than 250 + 1 = 251 or lesser than 250 - 1 = 249.
Since the normal distribution is symmetric, these probabilities are the same, so we can find one of them and multiply by 2.
Lesser than 249:
pvalue of Z when X = 249.
has a pvalue of 0.3707
0.3707*0.2 = 0.7414
74.14% probability that the volume of a single randomly chosen can differs from the target value by 1 ml or more.
c) What is the probability that the sample mean volume of a random sample of 6 cans differs from the target value by 1 ml or more?
Since more than 1 can, we use the Central Limit Theorem.
The probability follows the same logic as b.
By the Central Limit Theorem
has a pvalue of 0.2061
2*0.2061 = 0.4122
41.22% probability that the sample mean volume of a random sample of 6 cans differs from the target value by 1 ml or more