the study hours are 4, 10, 20, 24, 34, 36, 42 and the z scores are 2.5, negative 1.75, negative .5, 0, 1.25, 1.5, 2.25 and the percentile ranks are 10,25,35,40,50,75,95 what is the study hour with a z score of 1.75, what is the standard deviation of all the students, and what the is the study hour 2.5 standard deviations above the mean?
To find the study hour with a z-score of 1.75, we can use the formula:
Study hour = Mean + (z-score * Standard deviation)
Given that the z-score is 1.75, the mean and standard deviation are not provided directly. However, we can use the given data to calculate them.
Mean:
To find the mean, we add up all the study hours and divide by the number of data points:
Mean = (4 + 10 + 20 + 24 + 34 + 36 + 42) / 8 = 170 / 8 = 21.25 (rounded to two decimal places)
Standard Deviation:
To find the standard deviation, we need to calculate the variance first. The variance is the average of the squared differences between each study hour and the mean. Then, the standard deviation is the square root of the variance.
Step 1: Calculate the squared differences:
Subtract the mean from each study hour, square the result, and sum them all up:
(4 - 21.25)^2 + (10 - 21.25)^2 + (20 - 21.25)^2 + (24 - 21.25)^2 + (34 - 21.25)^2 + (36 - 21.25)^2 + (42 - 21.25)^2 = 2407.25
Step 2: Calculate the variance:
Divide the sum of squared differences by the number of data points:
Variance = 2407.25 / 8 = 300.91 (rounded to two decimal places)
Step 3: Calculate the standard deviation:
Take the square root of the variance:
Standard Deviation = √300.91 ≈ 17.35 (rounded to two decimal places)
Now, we can calculate the study hour with a z-score of 1.75:
Study hour = Mean + (z-score * Standard deviation)
Study hour = 21.25 + (1.75 * 17.35)
Study hour ≈ 53.31 (rounded to two decimal places)
Therefore, the study hour with a z-score of 1.75 is approximately 53.31.
To find the study hour that is 2.5 standard deviations above the mean, we can use the same formula:
Study hour = Mean + (z-score * Standard deviation)
Given that the z-score is 2.5:
Study hour = 21.25 + (2.5 * 17.35)
Study hour ≈ 63.38 (rounded to two decimal places)
Therefore, the study hour 2.5 standard deviations above the mean is approximately 63.38.
Finally, to calculate the standard deviation of all the students, we don't need the z-scores or percentile ranks. We can use the formula for the population standard deviation since we have the data for the entire population.
To find the standard deviation, we can go back to the variance calculation:
Standard Deviation = √Variance
Standard Deviation = √300.91 ≈ 17.35 (rounded to two decimal places)
Therefore, the standard deviation of all the students' study hours is approximately 17.35.