Question

according to the sleep in america—, americans get an average of 6 hours and 57 minutes of sleep per night. a survey of a class of 190 statistics students at a large university found that they averaged 7.1 hours of sleep the previous night, with a standard deviation of 1.95 hours.
(a) assume that the population average for adults is 6 hours and 57 minutes, or 6.95 hours of sleep per night, with a standard deviation of 2 hours. draw a picture similar to figure 19.3, illustrating how the rule for sample means would apply to sample means for random samples of 190 adults.
(b) would the mean of 7.1 hours of sleep obtained from the statistics students be a reasonable value to expect for the sample mean of a random sample of 190 adults? explain.
(c) can the sample taken in the statistics class be considered to be a representative sample of all adults? explain.

Explanation:

Step1: Calculate the standard error

The population mean $\mu = 6$ hours and $57$ minutes $= 6.95$ hours, population standard - deviation $\sigma = 2$ hours, and sample size $n = 190$. The standard error of the mean $SE=\frac{\sigma}{\sqrt{n}}=\frac{2}{\sqrt{190}}\approx\frac{2}{13.784}= 0.145$.

Step2: Find the intervals for 68 - 95 - 99.7 rule

For the 68 - 95 - 99.7 rule:

• For 68% confidence interval: $\mu\pm SE$, so $6.95\pm0.145$, which gives $(6.805,7.095)$.
• For 95% confidence interval: $\mu\pm2SE$, so $6.95\pm2\times0.145 = 6.95\pm0.29$, which gives $(6.66,7.24)$.
• For 99.7% confidence interval: $\mu\pm3SE$, so $6.95\pm3\times0.145 = 6.95\pm0.435$, which gives $(6.515,7.385)$.

(b)
The mean of 7.1 hours of sleep obtained from the statistics students:
Since $7.1$ is within the 95% confidence interval $(6.66,7.24)$ of the possible sample - means based on samples of size $n = 190$, it is a reasonable value to expect for the sample mean of a random sample of 190 adults. A mean of 7.1 hours is about 0.15 hours above the population mean of 6.95 hours, which is within the range of what is expected.

(c)
The sample taken in the statistics class cannot be considered a representative sample of all adults. College students are likely to be different from all adults in factors affecting sleep, such as age range, flexibility of schedules, perceived need and desire to stay up all night, and so on.

Answer:

(b) Yes, a mean of 7.1 hours is within the 95% confidence interval of the possible sample - means based on samples of size 190, so it is a reasonable value.
(c) No, college students are different from all adults in factors affecting sleep.