A sample of the number of study per week of college students is collected. Some of the sample results of the survey are given in the following table
Study Hr 4, 10, 20,24, 34, 36, 42
Z score -2.5,-1.75,-0.5,0, 1.25,1.5,2.25
P. Rank 10, 25, 35, 40, 50, 75 95
Q1- The mean study time for all the students
Q2- The median study time for all the student
Q3- The percent of college students who study between 20 and 36 hrs per week
Q4- The study time representing the 1st quartile
Q5- The Standard Deviation study time for all the students
Q5- The study time corresponding to a Z score of 1.75
Q1- The mean study time for all the students can be calculated by taking the average of all the study hours.
Mean = (4 + 10 + 20 + 24 + 34 + 36 + 42) / 7 = 170 / 7 = 24.29 (approximately)
Therefore, the mean study time for all the students is approximately 24.29 hours.
Q2- The median study time for all the students can be found by arranging the study hours in ascending order and finding the middle value.
Arranging the study hours in ascending order: 4, 10, 20, 24, 34, 36, 42
Since there are 7 data points, the middle value will be the (7+1)/2 = 8/2 = 4th value, which is 24.
Therefore, the median study time for all the students is 24 hours.
Q3- To find the percent of college students who study between 20 and 36 hrs per week, we need to calculate the cumulative probability between the corresponding percentile ranks.
The percent of college students who study between 20 and 36 hrs per week can be found by subtracting the cumulative probability at 20 hours from the cumulative probability at 36 hours.
Percent of college students = P.Rank (36) - P.Rank (20) = 50 - 35 = 15
Therefore, the percent of college students who study between 20 and 36 hrs per week is 15%.
Q4- The 1st quartile represents the 25th percentile (P.Rank = 25). To find the study time representing the 1st quartile, we need to look for the corresponding study hour at that percentile rank.
From the given table, at P.Rank = 25, the study hour is 10.
Therefore, the study time representing the 1st quartile is 10 hours.
Q5- The standard deviation of study time for all the students can be calculated by using the formula:
Standard Deviation = √[ Σ (Xi - X̄)^2 / N ]
where Σ represents the sum, Xi represents each study hour, X̄ represents the mean, and N represents the total number of data points.
Calculating the standard deviation:
Standard Deviation = √[ (4 - 24.29)^2 + (10 - 24.29)^2 + (20 - 24.29)^2 + (24 - 24.29)^2 + (34 - 24.29)^2 + (36 - 24.29)^2 + (42 - 24.29)^2 / 7 ]
Standard Deviation = √[ 404.2866 / 7 ]
Standard Deviation ≈ √57.7552 ≈ 7.60 (approximately)
Therefore, the standard deviation of study time for all the students is approximately 7.60 hours.
Q6- To find the study time corresponding to a Z-score of 1.75, we can use the formula:
Study Time = Z-score * Standard Deviation + Mean
Study Time = 1.75 * 7.60 + 24.29
Study Time ≈ 13.3 + 24.29
Study Time ≈ 37.59 (approximately)
Therefore, the study time corresponding to a Z score of 1.75 is approximately 37.59 hours.
To answer these questions, we will use the given sample data.
Q1- The mean study time for all the students:
The mean is calculated by summing up all the values in the sample and dividing it by the total number of values. In this case, the sum of the study hours is 4 + 10 + 20 + 24 + 34 + 36 + 42 = 170. Since there are 7 values in the sample, you divide the sum by 7. Therefore, the mean study time is 170/7 = 24.29 hours.
Q2- The median study time for all the students:
The median is the middle value when the data is arranged in ascending order. In this case, the data is already in ascending order: 4, 10, 20, 24, 34, 36, 42. The median is the value in the middle, which is 24. So, the median study time is 24 hours.
Q3- The percent of college students who study between 20 and 36 hours per week:
To find the percentage, we need to calculate the proportion of students who study between 20 and 36 hours and then multiply it by 100. Looking at the data, we can see that 4 students study between 20 and 36 hours per week (20, 24, 34, 36). Since the total sample size is 7, the proportion is 4/7 = 0.57. Multiplying it by 100 gives 57%. Therefore, 57% of college students study between 20 and 36 hours per week.
Q4- The study time representing the 1st quartile:
The quartiles divide the data into four equal parts. The first quartile is the median of the lower half of the data. Since the data is already in ascending order, the lower half is 4, 10, and 20. The median of these three numbers is 10. So, the study time representing the 1st quartile is 10 hours.
Q5- The standard deviation of study time for all the students:
The standard deviation measures the dispersion or variability of the data. To calculate standard deviation, you need to find the variance first. The variance is calculated by summing the squared differences between each data point and the mean, dividing it by the number of data points. Then, take the square root of the variance to find the standard deviation. Here is the step-by-step calculation:
- Calculate the difference for each data point from the mean: (-20.29, -14.29, -4.29, -0.29, 9.71, 11.71, 17.71)
- Square each difference: (414.84, 204.12, 18.36, 0.08, 94.44, 137.44, 313.84)
- Sum up the squared differences: 1183.12
- Divide the sum by the total number of data points (7): 1183.12/7 = 169.02
- Take the square root of the variance: √169.02 = 12.99
Therefore, the standard deviation of the study time for all the students is approximately 12.99 hours.
Q6- The study time corresponding to a Z score of 1.75:
The Z score gives the number of standard deviations a data point is away from the mean. To find the study time corresponding to a Z score of 1.75, we need to use the Z-score formula:
Z = (X - mean) / standard deviation
Solving for X (study time), we rearrange the formula as follows:
X = Z * standard deviation + mean
Substituting the given values, we get:
X = 1.75 * 12.99 + 24.29
X ≈ 47.73
Therefore, the study time corresponding to a Z score of 1.75 is approximately 47.73 hours.
We do not do your work for you. Once you have attempted to answer your questions, we will be happy to give you feedback on your work. Although it might require more time and effort, you will learn more if you do your own work. Isn't that why you go to school?
However, I will give you a start.
1. Z = 0 = mean
5. Z = (score-mean)/SD
Once you have found the mean, insert its value plus the Z and raw score for one student to solve for SD.
Use that data for 3 by finding table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the corresponding Z scores.