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Answer:
54.86% probability that the mean diameter of the sample shafts would differ from the population mean by more than 0.1 inches
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using 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 problem, we have that:
What is the probability that the mean diameter of the sample shafts would differ from the population mean by more than 0.1 inches
Lesser than 208 - 0.1 = 207.9 or greater than 208 + 0.1 = 208.1. Since the normal distribution is symmetric, these probabilities are equal, so we find one of them and multiply by 2.
Lesser than 207.9.
pvalue of Z when X = 207.9. So
By the Central Limit Theorem
has a pvalue of 0.2743
2*0.2743 = 0.5486
54.86% probability that the mean diameter of the sample shafts would differ from the population mean by more than 0.1 inches
At first glance, this problem might be overwhelming so try approaching it slowly by doing these steps:
Step 1: Start by understanding what the problem is really asking.
You might notice that the problem is just a percentage problem, with a twist! If you still don't see it, the problem is just asking what the base should be given a percentage and the amount.
Step 2: Recall the concepts relevant to the question.
We can identify a single formula that will be important in getting the answer:
Step 3: Apply the concept to the context of the problem.
You'll see that we already have the percent from the interest rate which is 0.02% and the amount from the below-minimum-balance fee which is $9.99.
There goes the little twist: you have to manipulate the equation to get the base. Dividing both sides of the equation by the percent we'll have:
Step 4: Evaluate to get the answer to the problem.
We simply substitute the given numbers and evaluate.
Answer: You need $49,950 in the account to cover the Below-Minimum-Balance Fee from your interest.
Step 1: Start by understanding what the problem is really asking.
You might notice that the problem is just a percentage problem, with a twist! If you still don't see it, the problem is just asking what the base should be given a percentage and the amount.
Step 2: Recall the concepts relevant to the question.
We can identify a single formula that will be important in getting the answer:
Step 3: Apply the concept to the context of the problem.
You'll see that we already have the percent from the interest rate which is 0.02% and the amount from the below-minimum-balance fee which is $9.99.
There goes the little twist: you have to manipulate the equation to get the base. Dividing both sides of the equation by the percent we'll have:
Step 4: Evaluate to get the answer to the problem.
We simply substitute the given numbers and evaluate.
Answer: You need $49,950 in the account to cover the Below-Minimum-Balance Fee from your interest.