Question

In order to encourage their students to read books, a primary school launched a one-year reading scheme. All students have 100 basic scores in the "reading report" at the beginning of the school year. When a student reads a book, the student will get 2 scores in the reading report. Students who get the top 10% scores at the end of the scheme will be issued a certificate. The number of books (X) read by a sample of 15 students are as follow: (a) Find the sample mean and the standard deviation of the number of books read from the above data. (b) Find the median and 90

th
percentile of the number of books read from the above data. (c) Use Y to denote the student's scores in the reading report and X to denote the number of books read by the student. Express Y in terms of X.

Answer 1

The sample mean and standard deviation of the number of books read are 76.67 and 25 respectively. The median and 90th percentile of the number of books read are 85 and 115 respectively. The scores in the reading report (Y) can be expressed in terms of the number of books read (X) as Y = 2X.

Step-by-step explanation:

(a) Sample Mean and Standard Deviation:

To find the sample mean, we need to add up all the values of X and divide it by the number of values. In this case, the values of X are:

50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120

Adding them up, we get:

50 + 55 + 60 + 65 + 70 + 75 + 80 + 85 + 90 + 95 + 100 + 105 + 110 + 115 + 120 = 1150

Dividing by the number of values (15), we get the sample mean:

1150 / 15 = 76.67

To find the standard deviation, we need to first find the variance. The variance is the average of the squared differences between each value of X and the sample mean. Using the formula for variance, we can calculate it as:

Variance = ((50 - 76.67)^2 + (55 - 76.67)^2 + ... + (120 - 76.67)^2) / 15 = 625.33

The standard deviation is the square root of the variance:

Standard Deviation = √625.33 ≈ 25

(b) Median and 90th Percentile:

To find the median, we need to arrange the values of X in ascending order:

50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120

Since we have an odd number of values (15), the median is the middle value, which in this case is the 8th value:

Median = 85

To find the 90th percentile, we need to calculate the index value. The index value is calculated using the formula:

Index = (Percentile / 100) * (n + 1)

where n is the number of values. In this case, Percentile = 90 and n = 15. Plugging in the values, we get:

Index = (90 / 100) * (15 + 1) = 13.6

Since the index is not a whole number, we need to round up to the next whole number to find the 90th percentile. This means that the 90th percentile is the value at the 14th index, which in this case is:

90th Percentile = 115

(c) Expressing Y in terms of X:

Since a student gets 2 scores in the reading report for each book read, we can express Y (scores in the reading report) in terms of X (number of books read) as:

Y = 2X