First you need to collect a small data set in the following

First you need to collect a small data set in the following way.

Record, to the nearest quarter of an hour, how many hours you slept in the past 24 hours.

8 Hours // Myself

Ask 3 more adults (in your family, in your classroom, at work, in the neighborhood, etc) how many hours of sleep they got in the past 24 hours and record their answers here to the nearest quarter of an hour.

7.5 Hours // Person 1

7.0 Hours // Person 2

6.5 Hours // Person 3

Use your data set and the results of the following survey to answer the questions that follow:

Questions:

1. What percent of adults from the survey slept longer than you did?

2. How long did the person who slept the least in your data set (other than you) sleep? What percent of adults from the survey slept less than this person?

3. How long did the person who slept the most in your data set (other than you) sleep? What percent of adults from the survey slept more than this person?

4. A colleague at your work came late to the meeting this morning. He said that he overslept. When you asked, he reported that he slept about 8.5 hours. What percent of adults from the survey slept less than this person?

5. Your boss, setting an example for the employees, said that he slept 6.75 hours last night to prepare for the meeting. What percent of adults from the survey slept longer than the boss?

6. What percent of data in the survey lies in the interval from 5.5 hours to 8.5 hours?

7. What interval about the mean includes 95% of the data in the survey?

8. What interval about the mean includes 50% of the data in the survey?

9. 30% of adults in the survey sleep less than how many hours of sleep?

10. 10% of adults in the survey sleep more than how many hours of sleep?

Solution

1) P(X > 8) = P(Z > (8 - 7)/1.25) = P(Z > 0.8)

= 0.2119 {From Standard normal table)

= 21.19%

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2) Person 3 slept the least in data set. Hours = 6.5

P(X < 6.5) = P(Z < (6.5 - 7)/1.25) = P(Z < -0.4)

= 0.3446

= 34.46%

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3) Person 1 slept for the most time = 7.5 hours

P(X > 7.5) = P(Z > (7.5 - 7)/1.25) = P(Z > 0.4)

= 0.3446

= 34.46%

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4) P(X < 8.5) = P(Z < (8.5 - 7)/1.25)

= P(Z < 1.2)

= 0.8849

= 88.49%

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5) P(X > 6,75) = P(Z > (6.75 - 7)/1.25)

= P(Z > -0.2)

= 0.5793

= 57.93%

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6) P( 5.5 < X < 8,5) = P( (5.5 - 7)/1.25 < Z < (8.5 - 7)/1.25)

= P( -1.2 < Z < 1.2)

= P(Z < 1.2) - P(Z < -1.2)

= 0.7699

76.99%

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7) P( -y < X < y) = 95%

From standard normal table, the value of standard normal variate = 1.96

=> Interval = (7 - 1.96*1.25 , 7 + 1.96*1.25)

= ( 4.55 , 9.45)

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8) P( -y < X < y) = 50%

From standard normal table, the value of standard normal variate = 0.6745

=> Interval = (7 - 0.6745*1.25 , 7 + 0.6745*1.25)

= ( 6.1569, 7.8431)

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9) let y be the required value

Now, P( X < y) = 0.3

From standard normal table, the value of standard normal variate = -0.5244

=> y = --0.5244*1.25 + 7

=> y = 6.3445 hrs

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10) let y be the required value

Now, P( X > y) = 0.1

From standard normal table, the value of standard normal variate = 1.282

=> y = 1.282*1.25 + 7

=> y = 8.6025 hrs

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