In a survey of sleeping habits, 8400 adults were selected randomly and contacted

Question

In a survey of sleeping habits, 8400 adults were selected randomly and contacted by telephone. Based on a medical association's recommended amount of sleep for adults, respondents were asked, “Do you typically sleep more than 6 hours during the night?” Of those surveyed, only 46% reported that they did. Suppose the researchers that conducted the survey described above decide to conduct it again but will now interview only 5000 people. In the first survey of 8400 people, the observed proportion was 46%. Which of the following is TRUE about the sample proportion for the second survey?

By the law of large numbers, it would again be 46%.

By the law of large numbers, the smaller (second) survey will certainly produce a sample proportion farther from the true population proportion than the larger (first) survey.T

The proportion computed from the sample of 5000 people would be more accurate because smaller samples tend to be more homogeneous than larger samples.

None of the above.

Explanation / Answer

Intially 46% of people are calculated from a sample of 8400 people

But now if the sample is 5000

a) By the law of large numbers, it to be 46%.

The above statement is false

Even if the population is homogenous there is always a chance of percentage deviation from the true popualtion

b) The smaller survey will give a sample proportion farther from the true population proportion

It is directly not true since the small population represents the larger populations

Deviation exists but not in larger amounts

c) The proportion from the sample of 5000 people would be more accurate than large sample

this is wrong by central limit theorem, as population increases the distribution assumes the normal form and the population will be more homogenous and accurate

d) None of the above

Hence the option D is the right

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