Population Case

Chapter 7, question 13:

I would rather have a random sample of 1000 people because it is more likely to follow a normal population curve and statistically, it would have a larger number of people who are taller than 65 inches.

As stated in the question, people in a large population average 60 inches tall. So, in a normal population, the height of people drawn would be a bell curve with 60 inches being in the middle of the curve. Then, we would get people who are taller or shorter than 60 inches, as you move away from the middle. The rules of the bell curve are that 68% of the population would be within 1 standard deviation of 60 inches, 95% would be within 2 standard deviations, and 99% would be within 3 standard deviations.

Now, with 100 people we would get a pretty good idea of the population but there could be some biases such as having kids in the sample or races of people who are statistically shorter. These examples would create bias.

Taking the 1000 people sample will create a more accurate representation of the population. It is still possible to have kids in this sample but the amount of people in the sample would be large enough to imitate a realistic population. To that end, I think it is better to choose the sample with 1000 people.

Now, the law of averages means that the outcomes of a random event will "even out" within a small sample. However, this does not take into consideration varies biases that within an experiment. Perfect example was mentioned above with the kids. If the sample was with smaller amount of people and many kids would be included, the height would definitely be smaller than 65 inches. However, the large sample is more likely to provide proper random heights.

Chapter 7, question 18:

The interpretation for the margin of error is not necessarily correct, and depends on the alpha.

The margin of error is the value to which the researcher is unsure of their results. Usually, the larger the margin of error, the less confident the research is of their results, thus they need to increase the spread. If they are more confident, then the margin of error would be smaller.

Here is an example:

we can be 95% sure that most workers there earn between $25 and $35 an hour, (30 plus or minus 5$)

we can be 90% sure that most workers there earn between $20 and $40 an hour (30 plus or minus 10$)

we can be 80% sure that most workers there earn between $15 and $45 an hour (30 plus or minus 15$)

The margin of error varies based on the alpha. The more confidence, the smaller the margin of error, whereas the less confidence, the higher the margin of error.

As the example illustrates above, the margin of error changes each time while trying to determine the salary of the workers so the interpretations for the margin of error is not necessarily right.

Chapter 7, question 23:

The sample must contain 10,000 people for such a poll: 1/(.01)2 = 1/ 0.0001 = 10,000

Chapter 8, question 6:

a) The statement would be that 95 out of 100 samples would lie between $300 and $700.

The standard deviation is $100 and our mean is $500, +/- 2 standard deviations. 95% of the values lie between +/- 2 standard deviations of the mean for a normal distribution. This is all based on the Rule of Thumb, also known as the 68-95-99.7 rule.

b) The statement would be that 16 people out of 100 to fall between $480 and $520

$480 to $520 is +/- 0.2 standard deviations of the mean.

P(Z < 0.2) - P(Z < -0.2) = 0.5793 - 0.4207 = 0.1584.