Question

(15.08) A government sample survey plans to measure the LDL (bad) cholesterol level of an SRS of men aged 20 to 34. The researchers will report the mean x¯¯¯x¯ from their sample as an estimate of the mean LDL cholesterol level μμ in this population. Step 1:
Explain to someone who knows no statistics what it means to say that x¯¯¯x¯ is an unbiased estimator of μμ

a.	x is systematically higher than or lower than μ.

b.	x is not systematically higher than or lower than μ.

Step 2:
The sample result x¯¯¯x¯ is an unbiased estimator of the population truth μμ no matter what size SRS the study uses. Explain to someone who knows no statistics why a large sample gives more trustworthy results than a small sample.

A.	With large samples, x is more likely to be further away from μ, because with a larger sample comes less information (and therefore more uncertainty).

B	With large samples, x is more likely to be further away from μ, because with a larger sample comes more information (and therefore less uncertainty).

C	With large samples, x is more likely to be close to μ, because with a larger sample comes less information (and therefore more uncertainty).

D	With large samples, x is more likely to be close to μ, because with a larger sample comes more information (and therefore less uncertainty).

Step 3:
Choose an SRS of 1000 men from this population.
Now what is the probability that x falls within ±3 mg/dL of μ? HINT: Find the standard deviation of x .
Use Table A or your calculator and give your answer to 3 decimal places.

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(15.10) Juan makes a measurement in a chemistry laboratory and records the result in his lab report. The standard deviation of students' lab measurements is σ = 10 milligrams. Juan repeats the measurement 4 times and records the mean x of his 4 measurements.


Step 1:
What is the standard deviation of Juan's mean result? (That is, if Juan kept on making 4 measurements and averaging them, what would be the standard deviation of all his x 's?) Give your answer as a whole number.

_____



Step 2:
How many times must Juan repeat the measurement to reduce the standard deviation of x to 2? (Give your answer as a whole number.)
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Step 3:
What is the advantage of reporting the average of several measurements rather than the result of a single measurement?

A	The average of several measurements is more likely to be close to the mean than the result of a single measurement.

B	Taking several measurements makes the standard deviation larger than that of a single measurement.

C	The average of several measurements gives an exact result.

5.

(15.20) Scores on the Critical Reading part of the SAT exam in a recent year were roughly Normal with mean 496 and standard deviation 115. You choose an SRS of 100 students and average their SAT Critical Reading scores. If you do this many times, the mean of the average scores you get will be close to

A	496.
B	496/100 = 4.96.
C	496/√100 = 49.6.

Answer 1

Step 1 : Option b is correct. x is not systematically higher than or lower than μ.

Step 2 : Option d is correct. With large samples, x is more likely to be close to μ, because with a larger sample comes more information (and therefore less uncertainty).

Problem 5.10

Step 1: σ = 10 milligrams

n = 4

standard deviation of Juan 's mean result = σ/sqrt(n) = 10/sqrt(4) = 5 mg/dl

Step 2 :

if we want to make standard deviation of mean = 2 mg/dl

σ/sqrt(n) = 2

sqrt(n) = 10/2 = 5

n = 25

Step 3 :

he average of several measurements is more likely to be close to the mean than the result of a single measurement. Option a is correct here.

Question 5

Here standard deviation = 115

Population mean= 496

Here option a is correct as the sample size is increased, it will get closer to population mean 496.