## A.6 Answer: TW 6 tutorial

### Answers for Sect. 6.2

Answers embedded. Some answers:

**1.** In order from left to right: \(70\); \(85\); \(100\); \(115\); \(130\).
**3.** \(30\) points above the mean.
**4.** Two standard deviations above the mean.
**5.** About \(2.5\)%.

A person with an IQ of \(85\) has an IQ that is **15** points **below** the mean, equivalent to **1** standard deviation(s) **below** the mean IQ.
Using the **68**--\(95\)--\(99.7\) rule, the proportion of the population with an IQ less than \(85\) is about **32**%.
In addition, the proportion of the population with an IQ **above** \(85\) is about **68**%.

### Answers for Sect. 6.3

- Answers vary.
- Answers vary.
- Answers vary.
- Answers vary. You probably cannot be very accurate using the \(68\)--\(95\)--\(99.7\) rule.
- Answers vary.
- Two standard deviations from the mean is \(2\times 6.7 = 13.4\), so \(95\)% of females aged \(18\) and over have a measured height between \(161.4 \pm13.4\) approximately, or from \(148.0\)cm to \(174.8\)cm.
- As follows:
- \(z = (171 - 161.4)/6.7 = 1.43\), so the probability is \(0.9236\) or about \(92\)%.
- So the odds are \(92.36/(100 - 92.36) = 12.1\).
- The answer is just \(1 - 0.9236 = 0.0764\) or about \(7.6\)%.
- The probability of
**over 171cm**is \(7.64\)%. - So the odds are \(7.64/(100 - 7.64)= 0.084\).
- \(z\)-scores are \(1.28\) and \(2.78\), so the answer is \(0.9973 - 0.8997 = 0.0976\), or about \(9.8\)%.
- So the odds are \(9.8/(100 - 9.8) = 0.11\).

- Use the Tables: \(z = -0.84\); then using unstandardising formula, the height is \(x = \mu + (z\times\sigma) = 161.4 + (-0.84\times6.7) = 155.772\), or about \(156\) cm.

### Answers for Sect. 6.4

Answers embedded.

Relative frequency. Relative frequency. Subjective. Relative frequency. Classical.

### Answers for Sect. 6.5

Answers implied by H5P.

The decision-making process begins with making an assumption about the population

**parameter**. This means we know what to**expect**from the sample**statistic**. We never know exactly what value of the statistic we will see in the sample, because of**sampling variation**. But we can have some of idea of what values are reasonable to expect. Then we take the**sample**(that is, we make the observations). Then we**compare**the sample statistic that we observed... to the sample statistic we expected. If what we observe is inconsistent with what was expected, then the the assumption is**unlikely to be**true. However, if what we observe is**consistent**with what was expected, then the the assumption is**probably**true.Step 1: Assumption about population parameter. Step 2: Expectation for sample statistic. Step 3: Observation of sample statistic. Decision: Consistent? Conclusion A: Yes, supports assumption. Conclusion B: No, doesn't support assumption.

### Answers for Sect. 6.6

- Depends on data.
- Looking close to normal, centred around \(0.5\)-ish.
- We'd have an approximate normal distribution with mean \(\mu_{\hat{p}} = 0.5\) and standard deviation \(\text{s.e.}(\hat{p}) = \sqrt{0.5 \times (1 - 0.5)/10} \approx 0.158\).
- Still normal; same mean; but standard deviation would be smaller: \(\text{s.e.}(\hat{p}) = \sqrt{0.5 \times (1 - 0.5)/50} \approx 0.071\).

### Answers for Sect. 6.7.1

- \(z = (20 - 0)/10 = 2\).
The area or probability to the
*right*is \(0.0228\), or about \(2.3\)%. - The probability the SOI exceeds \(20\): \(2.3\)%. So odds: \(2.3\div(100 - 2.3)\), or about \(0.024\).
- \(z = (-25 - 0)/10 = -2.5\).
The area to the
*left*is \(0.0062\), or about \(0.6\)%. - \(z = (-12 - 0)/10 = -1.2\).
The area to the
*right*is \(0.8849\), or about \(88.5\)%. - The two \(z\)-scores: \((-10 - 0)/10 = -1\) and \((20 - 0)/10 = 2\).
The area
*between*these is \(0.9772 - 0.1587 = 0.8185\), or about \(81.9\)%. - The \(z\) score corresponds to an area of \(0.80\) to the
*left*; from the tables, about \(z = 0.84\). This corresponds to an SOI of \(0 + (0.84\times 10)\), or an SOI of about \(8.4\). - The \(z\)-score is \(0.385\) from Appendix 3 in the textbook, remembering that the area to the
*left*would be \(0.650\) (draw a diagram!). So, the \(z\)-score is \(0.385\), so that the SOI values is \(x = \mu + (z\times\sigma) = 0 + (0.385\times 10) = 3.85\).