Suppose that the average weight of students in a statistics class is 75.5 kilograms with a standard deviation of 10.3 kilograms.
A. 45% of the student weights are above ____ kilograms?
B. What percent of students have weights greater than 80.1 kilograms?
C. What percent of students weight less than 60.1 kilograms?
D. What percent of students have weights between 73.3 kilograms and 77.4 kilograms?
E. Using the 68-95-99.7 rule, what is the lower bound for the weight of the top 2.5%?
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To answer these questions, we will use the concept of the standard normal distribution, also known as the Z-distribution. The Z-distribution assumes that the data follows a normal distribution with a mean of 0 and a standard deviation of 1. By standardizing the data using Z-scores, we can find the corresponding percentiles.
A. To find the weight above a certain value using percentiles, we need to find the Z-score associated with that value. The formula to calculate the Z-score is:
Z = (X - μ) / σ
where X is the value, μ is the mean, and σ is the standard deviation.
Given that the average weight (μ) is 75.5 kg and the standard deviation (σ) is 10.3 kg, we can calculate the Z-score for the weight above a certain value.
For Example, if we want to find the weight above X kilograms, we calculate the Z-score as follows:
Z = (X - 75.5) / 10.3
To find the weight above a certain percentile, we need to find the corresponding Z-score using a Z-table or statistical software.
B. To find the percent of students with weights greater than 80.1 kilograms, we need to find the area under the normal distribution curve to the right of that value. We can find the Z-score corresponding to 80.1 kg and then find the area to the right of that Z-score in the Z-table.
C. To find the percent of students with weights less than 60.1 kilograms, we need to find the area under the normal distribution curve to the left of that value. We can find the Z-score corresponding to 60.1 kg and then find the area to the left of that Z-score in the Z-table.
D. To find the percent of students with weights between 73.3 kg and 77.4 kg, we need to find the area under the normal distribution curve between these values. We can find the Z-scores corresponding to these values and then find the area between these Z-scores in the Z-table.
E. The 68-95-99.7 rule states that in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. By using this rule, we can find the lower bound for the weight of the top 2.5%.
We can calculate the Z-score corresponding to the top 2.5% using the Z-table or statistical software. Then, we can find the corresponding weight by using the formula:
X = Z * σ + μ
where X is the weight, Z is the Z-score, σ is the standard deviation, and μ is the mean.