A painter has 24 foot ladder that he is using to paint a house. For safety reasons, the ladder must be placed at least 8 feet fr
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Answer:
0.3821 = 38.21% probability that Ricardo makes a higher proportion of putts than Tammy.
Step-by-step explanation:
To solve this question, we need to understand the normal distribution, the central limit theorem, and subtraction of normal variables.
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.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean and standard deviation
Subtraction of normal variables:
When we subtract normal variables, the mean of the subtraction will be the subtraction of the means, while the standard deviation will be the square root of the sum of the variances.
Ricardo makes 47% of his putts, and attempts 25 putts.
By the Central Limit Theorem, we have that:
Tammy makes 51% of her putts, and attempts 30 putts.
By the Central Limit Theorem, we have that:
What is the probability that Ricardo makes a higher proportion of putts than Tammy?
This is the probability that the subtraction of R by T is larger than 0. The mean and standard deviation of this distribution are, respectively:
This probability is 1 subtracted by the pvalue of Z when X = 0. So
By the Central Limit Theorem
has a pvalue of 0.6179
1 - 0.6179 = 0.3821
0.3821 = 38.21% probability that Ricardo makes a higher proportion of putts than Tammy.