Assume that a population is normally distributed with a mean of 100 and a standard deviation of 15. Would it be unusual for the mean of a sample of 3 to be 115 or more? Why or why not?
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To determine whether it would be unusual for the mean of a sample of 3 to be 115 or more, we can calculate the standard error of the mean and then use z-scores to determine the probability.
The standard error of the mean (SE) is calculated by dividing the standard deviation by the square root of the sample size. In this case, the standard deviation is 15 and the sample size is 3:
SE = 15 / √3 ≈ 8.66
Next, we can calculate the z-score using the formula:
z = (sample mean - population mean) / SE
z = (115 - 100) / 8.66 ≈ 1.73
To find the probability associated with this z-score, we can refer to a standard normal distribution table or use statistical software.
From the standard normal distribution table, we find that the probability of obtaining a z-score of 1.73 or above is approximately 0.0416, which corresponds to 4.16%.
Therefore, it would be considered unusual for the mean of a sample of 3 to be 115 or more, as there is only a 4.16% chance of this occurring based on the given population parameters.
To determine whether it would be unusual for the mean of a sample of 3 to be 115 or more, we can use the concept of the sampling distribution of the mean.
The sampling distribution of the mean refers to the distribution of sample means that we would obtain if we repeatedly took samples of a fixed size from the same population.
In this case, we have a normally distributed population with a mean of 100 and a standard deviation of 15.
The formula to calculate the standard deviation of the sampling distribution of the mean, also known as the standard error, is given by:
Standard Error = Population Standard Deviation / √(Sample Size)
Applying this formula to our situation, we have:
Standard Error = 15 / √(3) = 8.66
According to the Central Limit Theorem, when the sample size is large (typically larger than 30), the sampling distribution of the mean tends to follow a normal distribution, regardless of the shape of the population distribution.
Since our sample size is only 3, it is relatively small. When the sample size is small, the shape of the population distribution starts to affect the shape of the sampling distribution of the mean.
Therefore, to determine whether it would be unusual for the mean of a sample of 3 to be 115 or more, we need to consider the z-score associated with a mean of 115 in the sampling distribution of the mean.
The z-score measures the number of standard errors a given value is from the mean. The formula for calculating the z-score is:
z = (Sample Mean - Population Mean) / Standard Error
Applying this formula to our situation, we have:
z = (115 - 100) / 8.66 = 1.73
To interpret the z-score, we can refer to a standard normal distribution table or use a calculator to find the corresponding percentile.
According to a standard normal distribution table, a z-score of 1.73 corresponds to a cumulative probability of approximately 0.9582 or 95.82%.
This means that a sample mean of 115 or more would be within the top 4.18% (100% - 95.82%) of the sampling distribution, which can be considered unusual based on conventional criteria.
Therefore, it would be considered unusual for the mean of a sample of 3 to be 115 or more, given the characteristics of the population distribution.
Note: It's important to understand that the term "unusual" is subjective and may vary depending on the specific context or criteria used. In statistics, common thresholds for defining unusual values are often based on a significance level of 0.05 (5%) or 0.01 (1%).
Use z-scores:
z = (x - mean)/(sd/√n)
x = 115
mean = 100
sd = 15
n = 3
Once you have z, use a z-table to determine probability. If the probability is low, then it would be unusual for the mean of a sample of 3 to be 115 or more.
I hope this will help.