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Answer :
Answer:
0.962 = 96.2% probability that a simple random sample of 100 adult males from this county has a mean weight between 172 and 188 lbs.
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 normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
The distribution of weights of adult males in a certain county is strongly right-skewed with a mean weight of 185 lbs and standard deviation 16 lbs.
This means that [tex]\mu = 185, \sigma = 16[/tex]
Sample of 100:
This means that [tex]n = 100, s = \frac{16}{\sqrt{100}} = 1.6[/tex]
What is the probability that a simple random sample of 100 adult males from this county has a mean weight between 172 and 188 lbs?
This is the p-value of Z when X = 188 subtracted by the p-value of Z when X = 172. So
X = 188
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{188 - 185}{1.6}[/tex]
[tex]Z = 1.875[/tex]
[tex]Z = 1.875[/tex] has a p-value of 0.9620
X = 172
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{172 - 185}{1.6}[/tex]
[tex]Z = -8.125[/tex]
[tex]Z = -8.125[/tex] has a p-value of 0.
0.9620 - 0 = 0.962
0.962 = 96.2% probability that a simple random sample of 100 adult males from this county has a mean weight between 172 and 188 lbs.
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The probability that a simple random sample of 100 adult males has a mean weight between 172 and 188 lbs can be calculated using the central limit theorem and z-scores. The standard error for the sampling distribution is 1.6 lbs, and we can find the probability by determining the z-scores for the weights and finding the corresponding probability range from the standard normal distribution.
The distribution of weights of adult males in a certain county is strongly right-skewed with a mean weight of 185 lbs and standard deviation 16 lbs. Students need to calculate the probability that a simple random sample of 100 adult males from this county has a mean weight between 172 and 188 lbs. Since the sample size is large (n=100), we can use the central limit theorem which states that the sampling distribution of the sample mean will be approximately normally distributed, even if the population distribution is not, provided the sample size is large enough.
First, we need to calculate the standard deviation of the sampling distribution, which is also known as the standard error (SE). The formula for the SE is:
SE = σ/√n
Where σ is the population standard deviation and n is the sample size. Substituting the given values:
SE = 16 /√(100) = 16 / 10 = 1.6 lbs
Next, we find the z-scores for the weights 172 and 188 using the formula:
Z = (X - μ)/SE
Where X is the value for which we are finding the z-score, \/mu is the population mean, and SE is the standard error. For X = 172 lbs and X = 188 lbs, the z-scores are:
[tex]Z_{172}[/tex] = (172 - 185) / 1.6 = -13 / 1.6 = -8.125
[tex]Z_{188}[/tex] = (188 - 185) / 1.6 = 3 / 1.6 = 1.875
Using standard normal distribution tables or software, we can find the probabilities corresponding to these z-scores and then find the probability that the mean weight lies between them. The probability of Z being between -8.125 and 1.875 will provide the final answer.
