Question

The results of a national survey showed that on average, adults sleep 6.6 hours per night. Suppose that the standard deviation is 1 hour and that the number of hours of sleep follows a bell-shaped distribution. If needed, round your answers to two decimal digits. If your answer is negative use "minus sign". (a) Use the empirical rule to calculate the percentage of individuals who sleep between 4.6 and 8.6 hours per day. Enter your answer as a percentage. (b) What is the z-value for an adult who sleeps 8 hours per night? (c) What is the z-value for an adult who sleeps 6 hours per night?

Answer 1

u= average = 6.6

0 = standard deviation = 1

Empirical rule

1) Approrimately 68% of the data will be between u-o and u to

We plug the value of average and standard deviation

Approximately 68% of the data will be between p - 0 = 6.6 – 1 = 5.6 and uto= 6.6+1 = 7.6

Approximately 68% of the data will be between 5.6 and 7.6

.

2) Approximately 95% of the data will be between u-2*0 and u +2*0

We plug the value of the average and standard deviation.

Approximately 95% of the data will be between u-2*0 = 6.6 – 2*1 = 6.6 – 2 = 4.6 and u +2*0= 6.6+2*1 = 6.6 +2 = 8.6

Approximately 95% of the data will be between 4.6 and 8.6

.

3) Approximately 99.7% of the data will be between u-3*o and u+3*0

We plug the value of the average and standard deviation.

Approximately 99.7% of the data will be between u-3*0 = 6.6–3*1 = 6.6 – 3 = 3.6 and u +3* = 6.6+3*1= 6.6+3 = 9.6

Approximately 99.7% of the data will be between 3.6 and 9.6

Answer for # a part

Approximately 95% of the data will be between 4.6 and 8.6

We get the answer as 95 % for # a part

#b part

x: number of hours sleeps per night in b part = 8

u= average = 6.6

0 = standard deviation = 1

Formula for Z score

- Z score = -

Z score = 8 – 6.6 -= 1.4

We round the z score value up to 2 decimal place

Z score = 1.40

Final answer for b part

Z score = 1.40

# c part

x: number of hours sleeps per night in c part = 6

u= average = 6.6

0 = standard deviation = 1

Formula for Z score

- Z score = -

Z score = 6 – 6.6 - = -0.6

We round the z score value up to 2 decimal place

Z score = -0.60

Final answer for c part

Z score = - 0.60

I hope this will help you :)