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Answer:
a) 10.38% probability that the sample mean will be more than 59 pounds.
b) 67.72% probability that the sample mean will be more than 56 pounds.
c) 22.10% probability that the sample mean will be between 56 and 57 pounds.
d) 1.46% probability that the sample mean will be less than 53 pounds.
e) 0% probability that the sample mean will be less than 49 pounds.
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 question, we have that:
a. More than 59 pounds
This is 1 subtracted by the pvalue of Z when X = 59. So
By the Central Limit Theorem
has a pvalue of 0.8962.
1 - 0.8962 = 0.1038
10.38% probability that the sample mean will be more than 59 pounds.
b. More than 56 pounds
This is 1 subtracted by the pvalue of Z when X = 56. So
has a pvalue of 0.3228.
1 - 0.3228 = 0.6772
67.72% probability that the sample mean will be more than 56 pounds.
c. Between 56 and 57 pounds
This is the pvalue of Z when X = 57 subtracted by the pvalue of Z when X = 56. So
X = 57
has a pvalue of 0.5438
X = 56
has a pvalue of 0.3228.
0.5438 - 0.3228 = 0.2210
22.10% probability that the sample mean will be between 56 and 57 pounds.
d. Less than 53 pounds
This is the pvalue of Z when X = 53.
has a pvalue of 0.0146
1.46% probability that the sample mean will be less than 53 pounds.
e. Less than 49 pounds
This is the pvalue of Z when X = 49.
has a pvalue of 0.
0% probability that the sample mean will be less than 49 pounds.