Chapter 7: Point Estimation and Sampling Distributions

1. If N=12 and n=3, what is the probability of any given simple random sample?

1. 0.001262626
2. 0.015151520
3. 0.004545455 X
4. 0.002020202

Solution:

> 1 / choose(12, 3)

[1] 0.004545455

2.  If N= 15 and n= 3, how many unique simple random samples are possible?

1. 1365
2. 105
3. 3003
4. 455 X

Solution:

> choose(15, 3)

[1] 455

3.   A population has a mean of m=100 and a standard deviation of s=15. If we draw a simple random sample of size n=36, what is the probability that the sample mean x̅ will be less than 105?  That is, what is p(x̅ < 105)?

1. 0.9901847
2. 0.9087888
3. 0.9772499 X
4. 0.9522096

Solution:

> pnorm((105 - 100) / (15 / sqrt(36)))

[1] 0.9772499

4.  A population has a mean of m=100 and a standard deviation of s=15. If we draw a simple random sample of size n=36, what is the probability that the sample mean x̅ will be greater than 95 but less than 97?  That is, what is p(95 < x̅ < 97)?

1. 0.1108649
2. 0.1206442
3. 0.07094133
4. 0.09231954 X

Solution:

> pnorm((97 - 100) / (15 / sqrt(36))) - pnorm((95 - 100) / (15 / sqrt(36)))

[1] 0.09231954

5.  A population has a mean of m=100 and a standard deviation of s=15. If we draw a simple random sample of size n=36, what is the probability that the sample mean x̅ will be greater than 98?  That is, what is p(x̅ > 98)?

1. 0.8246761
2. 0.7881446 X
3. 0.7030986
4. 0.7475075

Solution:

> pnorm((98 - 100) / (15 /sqrt(36)), lower.tail = FALSE)

[1] 0.7881446

6.  A population has a mean of m=100 and a standard deviation of s=15. If we draw a simple random sample of size n=81, what is the probability that the sample mean x̅ will be greater than 98?  That is, what is p(x̅ > 98)?

1. 0.9087888
2. 0.8849303 X
3. 0.8569388
4. 0.8271107

Solution:

> pnorm((98 - 100) / (15 /sqrt(81)), lower.tail = FALSE)

[1] 0.8849303

7.  A population has a proportion of p=0.45. If a simple random sample of 100 is drawn, what is the probability the sample population  is less than 0.47? That is, what is p( < 0.47)?

Solution:

> pnorm(0.47, 0.45, sqrt(0.45 * 0.55 / 100))

[1] 0.6561636

8.  A population has a proportion of p=0.45. If a simple random sample of 100 is drawn, what is the probability the sample population  is greater than 0.46 but less than 0.48? That is, what is p(0.46 <  < 0.48)?

1. 0.1470995 X
2. 0.1589428
3. 0.1345592
4. 0.1075812

Solution:

> pnorm(0.48, 0.45, sqrt(0.45 * 0.55 / 100)) - pnorm(0.46, 0.45, sqrt(0.45 * 0.55 / 100))

[1] 0.1470995

9.  A population has a proportion of p=0.45. If a simple random sample of 100 is drawn, what is the probability the sample population   is greater than 0.48? That is, what is p( > 0.48)?

Solution:

> pnorm(0.48, 0.45, sqrt(0.45 * 0.55 / 100), lower.tail = FALSE)

[1] 0.2732468

10.  A population has a proportion of p=0.45. If a simple random sample of 625 is drawn, what is the probability the sample population  is greater than 0.48? That is, what is p( > 0.48)?

1. 0.08056247
2. 0.19747690
3. 0.06583401 X
4. 0.03522021

Solution:

> pnorm(0.48, 0.45, sqrt(0.45 * 0.55 / 625), lower.tail = FALSE)

[1] 0.06583401