Math

Question50 students answered a math question. Given the time intervals and frequencies, estimate the mean time taken. Use intervals: 10<t2010<t \leqslant 20, 20<t2520<t \leqslant 25, 25<t3025<t \leqslant 30, 30<t5030<t \leqslant 50, 50<t8050<t \leqslant 80. Frequencies: 2, 8, 12, 16, 12.

Studdy Solution

STEP 1

Assumptions1. The table provided represents the distribution of time taken by50 students to answer a mathematics question. . The time intervals are given in seconds.
3. The frequency represents the number of students who took a certain amount of time to answer the question.
4. The mean time is estimated by taking the midpoint of each time interval, multiplying it by the frequency, summing these products, and then dividing by the total number of students.

STEP 2

First, we need to find the midpoint of each time interval. The midpoint can be calculated by adding the lower limit and the upper limit of the interval and dividing by2.
Midpoint=Lowerlimit+Upperlimit2Midpoint = \frac{{Lower\, limit + Upper\, limit}}{2}

STEP 3

Calculate the midpoints for each time interval.
For the first interval, 10<t2010<t \leqslant20, the midpoint isMidpoint=10+202Midpoint = \frac{{10 +20}}{2}

STEP 4

Calculate the value of the midpoint.
Midpoint=10+202=15Midpoint = \frac{{10 +20}}{2} =15

STEP 5

Repeat the process for the remaining intervals.
For the second interval, 20<t2520<t \leqslant25, the midpoint isMidpoint=20+252Midpoint = \frac{{20 +25}}{2}

STEP 6

Calculate the value of the midpoint.
Midpoint=20+252=22.5Midpoint = \frac{{20 +25}}{2} =22.5

STEP 7

Continue with the remaining intervals.
For the third interval, 25<t3025<t \leqslant30, the midpoint isMidpoint=25+302Midpoint = \frac{{25 +30}}{2}

STEP 8

Calculate the value of the midpoint.
Midpoint=25+302=27.5Midpoint = \frac{{25 +30}}{2} =27.5

STEP 9

For the fourth interval, 30<t5030<t \leqslant50, the midpoint isMidpoint=30+502Midpoint = \frac{{30 +50}}{2}

STEP 10

Calculate the value of the midpoint.
Midpoint=30+502=40Midpoint = \frac{{30 +50}}{2} =40

STEP 11

For the fifth interval, 50<t8050<t \leqslant80, the midpoint isMidpoint=50+80Midpoint = \frac{{50 +80}}{}

STEP 12

Calculate the value of the midpoint.
Midpoint=50+802=65Midpoint = \frac{{50 +80}}{2} =65

STEP 13

Next, we multiply each midpoint by its corresponding frequency to get the total time for each interval.
For the first interval, the total time isTotaltime=MidpointtimesFrequencyTotal\, time = Midpoint \\times FrequencyTotaltime=15times2Total\, time =15 \\times2

STEP 14

Calculate the total time for the first interval.
Totaltime=times2=30Total\, time = \\times2 =30

STEP 15

Repeat the process for the remaining intervals.
For the second interval, the total time isTotaltime=22.5times8Total\, time =22.5 \\times8

STEP 16

Calculate the total time for the second interval.
Totaltime=22.5times8=180Total\, time =22.5 \\times8 =180

STEP 17

Continue with the remaining intervals.
For the third interval, the total time isTotaltime=27.5times12Total\, time =27.5 \\times12

STEP 18

Calculate the total time for the third interval.
Totaltime=27.5times12=330Total\, time =27.5 \\times12 =330

STEP 19

For the fourth interval, the total time isTotaltime=40times16Total\, time =40 \\times16

STEP 20

Calculate the total time for the fourth interval.
Totaltime=40times16=640Total\, time =40 \\times16 =640

STEP 21

For the fifth interval, the total time isTotaltime=65times12Total\, time =65 \\times12

STEP 22

Calculate the total time for the fifth interval.
Totaltime=65times12=780Total\, time =65 \\times12 =780

STEP 23

Now, add up all the total times to get the overall total time.
Overalltotaltime=Totaltimeforfirstinterval+Totaltimeforsecondinterval+Totaltimeforthirdinterval+Totaltimeforfourthinterval+TotaltimeforfifthintervalOverall\, total\, time = Total\, time\, for\, first\, interval + Total\, time\, for\, second\, interval + Total\, time\, for\, third\, interval + Total\, time\, for\, fourth\, interval + Total\, time\, for\, fifth\, intervalOveralltotaltime=30+180+330+640+780Overall\, total\, time =30 +180 +330 +640 +780

STEP 24

Calculate the overall total time.
Overalltotaltime=30+180+330+640+780=1960Overall\, total\, time =30 +180 +330 +640 +780 =1960

STEP 25

Finally, we can calculate the estimated mean time by dividing the overall total time by the total number of students (50).
Estimatedmeantime=OveralltotaltimeTotalnumberofstudentsEstimated\, mean\, time = \frac{{Overall\, total\, time}}{Total\, number\, of\, students}Estimatedmeantime=196050Estimated\, mean\, time = \frac{{1960}}{50}

STEP 26

Calculate the estimated mean time.
Estimatedmeantime=196050=39.Estimated\, mean\, time = \frac{{1960}}{50} =39.The estimated mean time taken by each student to answer the question is39. seconds.

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