Question

7 points according to the documents, 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 1.85 hours.
(3) 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.
(4) 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.
(5) can this sample be considered to be representative of all adults? explain.

Explanation:

Step1: Identify key - values

Population mean $\mu = 6.95$ hours, population standard deviation $\sigma = 2$ hours, sample size $n = 190$.

Step2: Calculate standard error

The standard error of the mean $SE=\frac{\sigma}{\sqrt{n}}=\frac{2}{\sqrt{190}}\approx\frac{2}{13.784}\approx0.145$.

Step3: Analyze the sample - mean value

The sample - mean from students is $7.1$ hours. To check if it's a reasonable value for the sample mean of adults, we can use the empirical rule for the sampling distribution of the sample mean (which is approximately normal since $n = 190$ is large). According to the empirical rule, about 95% of the sample means lie within $\mu\pm2SE$. $\mu - 2SE=6.95-2\times0.145 = 6.95 - 0.29=6.66$ and $\mu + 2SE=6.95 + 0.29 = 7.24$. Since $7.1$ is within the interval $(6.66,7.24)$, it is a reasonable value for the sample mean of a random sample of 190 adults.

Step4: Consider representativeness

College students may not be representative of all adults because they may have different factors affecting sleep such as age range, flexibility of schedule, perceived need and desire to stay up all - night, etc.

Answer:

(b) Yes, the mean of 7.1 hours is a reasonable value to expect for the sample mean of a random sample of 190 adults because it lies within the interval of values (6.66, 7.24) which contains about 95% of the possible sample means. However, (c) No, the sample of statistics students is not likely to be a representative sample of all adults because college students have different factors affecting sleep compared to the general adult population.