Question

More than 100 million people around the world are not getting enough sleep; the average adult needs between 7.5 and 8 hours of sleep per night. College students are particularly at risk of not getting enough shut-eye. A recent survey of several thousand college students indicated that the total hours of sleep time per night, denoted by the random variable X, can be approximated by a normal model with E(X) = 6.94 hours and SD(X) = 1.12 hours. Find the probability that the hours of sleep per night for a random sample of 4 college students has a mean x bar between 6.6 and 6.93. (use 4 decimal places in your answer)

True or false?

Answer 1

The probability that the hours of sleep per night for a random sample of 4 college students has a mean x bar between 6.6 and 6.93 is 0.2209.

What is probability?

It is defined as the ratio of the number of favorable outcomes to the total number of outcomes, in other words, the probability is the number that shows the happening of the event.

We have:
SD(X) = 1.12 hours

Number of samples N = 4

Mean = 6.94

SD = 1.12/√4

SD = 0.56

Z = (X - mean)/SD = (X - 6.94)/0.56

$\rm P(6.6 < X < 6.93)=P(X < 6.93)-P(X < 6.6)$

$\rm = P(\dfrac{X-6.94}{0.56} < \dfrac{6.93-6.94}{0.56}) - P(\dfrac{X-6.94}{0.56} < \dfrac{6.6-6.94}{0.56})$

= P(Z < -0.0178) - P(Z < -0.607)

= 0.4929 -  0.27193

= 0.2209

Thus, the probability that the hours of sleep per night for a random sample of 4 college students has a mean x bar between 6.6 and 6.93 is 0.2209.

Learn more about the probability here:

brainly.com/question/11234923

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