04a Stats 1: January 2009

Monday, January 19, 2009

Chapter 11 - Video tutorials index

Video tutorial 11.1 Correlation

This recording discusses the concept of correlation. Advice is given with regard to producing a scatter diagram as well as appropriate comments to make. Computation of the sample correlation coefficient is also covered. Duration: 12 minutes 10 seconds.

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Video tutorial 11.2 Simple linear regression

The regression of a dependent variable, y, on a single independent variable, x, is explained. Estimation of the line of best fit to determine optimal point estimates of the intercept and slope parameters is discussed. Application of the estimated model for prediction purposes and a caveat concerning exctrapolation are also covered. Duration: 13 minutes 18 seconds.

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Video tutorial 11.3 Empirical regression example

Question: From n pairs of values (x_i, y_i), i = 1, 2, ... n, the following quantities are calculated:

n = 20,	Σx_i = 400,	Σy_i = 220,
Σx²_i = 8800,	Σx_iy_i = 4300,	Σy²_i = 2620.

Find the linear regression equations of y on x and x on y.

Duration: 10 minutes 43 seconds.

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Video tutorial 9.1 Essay question tips

The Statistics 1 examination frequently assesses a candidate’s knowledge of sampling. This recording offers some helpful guidance to the main issues examiners are likely to be looking for in a typical response. Duration: 10 minutes 5 seconds.

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Chapter 8 - Video tutorials index

Video tutorial 8.1 Titanic contingency table example

Question: The table below summarises the fate of the passengers and the crew when the Titanic sank on Monday, 15 April 1912. Test the hypothesis of independence between the row variable and the column variable in the table, and interpret your findings.

			Gender/Age	Category
		Men	Women	Boys	Girls
Fate	Survived	332	318	29	27
	Died	1360	104	35	18

Duration 36 minutes, 23 seconds.

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Chapter 7 - Video tutorials index

Video tutorial 7.1 Principles of hypothesis testing

Similar to Recording 6.1 for confidence intervals, this video explains the mechanics of hypothesis testing. Formation of the hypotheses is discussed along with the concept of a test statistic and its associated distribution. Specification of a significance level, α, is covered and its role in determining critical values in order to define an appropriate critical/rejection region. Interpretation of how to draw conclusions from comparing an observed test statistic value with the critical values is also covered. Duration: 40 minutes 47 seconds.

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Video tutorial 7.2 Test statistic scenarios

This recording is the hypothesis testing equivalent of Recordings 6.2 and 6.3. All the test statistics associated with testing (i) single population means (ii) differences between population means (iii) single population proportions and (iv) differences between population proportions are covered. Just as with confidence intervals, unknown population variances must be treated appropriately when testing means. Duration: 23 minutes 23 seconds.

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Video tutorial 7.3 Testing a single population mean

Question: It may be assumed that the length of nails produced by a particular machine is a normally distributed random variable, with standard deviation 0.02 cm. The lengths of a sample of 5 nails are 1.14, 1.15, 1.14, 1.12 and 1.16 cm. Test, at the 5% significance level, the hypothesis that the machine produces nails with a mean length 1.12 cm. Duration: 9 minutes 51 seconds.

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Video tutorial 7.4 Testing the difference between population means

Question: The mean reaction times, in hundredths of a second, of two groups of subjects to a flashing-light stimulus are measured. The first group, denoted by subscript 1, comprised individuals who were new to the experiment while the subjects in the second group, denoted by subscript 2, had previously taken part. Summary statistics for the two samples are given below.

₁= 3.0	s²₁ = 0.064	n₁ = 10
₂= 2.7	s²₂ = 0.031	n₂ = 8

Test if experience has had an effect on the mean response time. Duration: 15 minutes 25 seconds.

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Video tutorial 7.5 Testing a single population proportion

Question: An accounting firm wishes to test the claim that no more than 1% of a large number of transactions contains errors. In order to test this claim, they examine a random sample of 144 transactions and find that exactly 3 of these are in error. What conclusion should the firm draw? Duration: 11 minutes 0 seconds.

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Video tutorial 7.6 Testing the difference between population proportions

Question: A market research company has conducted a survey of adults in two large towns, either side of an international border, in order to judge attitudes towards a controversial internationally broadcast celebrity television programme. The following table shows some of the information obtained by the survey:

	Town A	Town B
Sample size	40	40
Sample number approving of the programme	24	22

Conduct a formal hypothesis test, at the 5% significance level, of the claim that the population proportions approving the programme in the two towns are equal. Duration: 12 minutes 11 seconds.

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Chapter 6 - Video tutorials index

Video tutorial 6.1 Principles of confidence intervals

The key ingredients to confidence intervals are outlined as well as their interpretation as a indicator to the level of uncertainty attached to a point estimate. A generic template is offered which can subsequently be applied to the various scenarios encountered in Statistics 1. Duration: 11 minutes 50 seconds.

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Video tutorial 6.2 Review of confidence interval scenarios: single parameters

Building on the generic template given in Video tutorial 6.1, the different scenarios of single point estimates which appear in the Statistics 1 subject guide are reviewed. These comprise confidence intervals estimating a single population mean, μ, and a single population proportion, π. For the former case involving means, consideration must be given to whether the population variance, σ ², can be assumed to be known and, if not, the mechanism for dealing with this is discussed. Duration: 15 minutes 26 seconds.

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Video tutorial 6.3 Review of confidence interval scenarios: differences between two parameters

This Video tutorial extends Recording 6.2 to consider constructing confidence intervals for differences between two population means, μ₁ - μ₂, and the difference between population proportions, π₁- π₂. Again for the mean case, attention must be paid to whether the two population variances, σ²₁ and σ²₂ respectively, are known. If unknown, estimating the variances individually and producing a pooled estimate are considered. Duration: 19 minutes 27 seconds.

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Video tutorial 6.4 Computing minimum sample size to achieve a given confidence interval width

Question: James thought the population proportion of the electorate who intended to vote for the incumbent government in the forthcoming election was 34%. To investigate this hunch he decides to estimate this with a sample proportion. He sets a tolerance limit of 0.10 with 90% confidence. What is the minimum sample size required? Duration: 11 minutes 12 seconds.

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Video tutorial 6.5 Confidence interval for the difference in proportions

Question: A researcher was investigating computer usage among students at a particular university. 200 undergraduates and 100 postgraduates were chosen at random and asked if they owned a laptop. It was found that 81 of the undergraduates and 63 of the postgraduates did. Find a 95% confidence interval for the difference in the proportion of undergraduates and postgraduates who own a laptop. Duration: 9 minutes 34 seconds.

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Video tutorial 6.6 Confidence interval for a population mean, variance unknown

Question: A random sample of 8 students was taken and their scores in a statistics paper recorded. The sample mean was calculated to be

= 71.2$ and the sample variance, s², was 4.9. Compute a 99% confidence interval for the mean statistics score for all students taking the course. How can the interval width be made narrower? Why? Duration: 11 minutes 34 seconds.

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Video tutorial 6.7 Confidence interval for the difference in population means, variances unknown

Random samples are taken from two populations with distribution N(μ_X,σ²) and $N(μ_Y, σ²) (i.e. their variances are the same). The summary statistics for the two samples are as follows:

	Sample size, n	Sample mean, m	Sample Variance s²
x-data	19	7.0	1.69
y-data	25	5.1	2.56

Compute a 95% confidence interval for the difference μ_X - μ_Y between the two population means. Duration: 13 minutes 5 seconds.

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Chapter 5 - Video tutorials index

Video tutorial 5.1 Introduction to the Normal distribution and standardisation

The Normal distribution is explained and it is noted that there exists an infinite number of combinations for the pair of parameter values for the mean, μ, and the variance, σ2. Since we are often interested in finding probabilities associated with Normal distributions, the reason behind the production of statistical tables for the Standard Normal distribution is discussed. Consequently the important technique of standardisation is discussed to convert a non-standard Normal variable into a standard Normal one, hence allowing consultation of the statistical tables. Duration: 14 minutes 4 seconds.

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Video tutorial 5.2 Calculation of Normal probabilities

This recording applies the standardisation technique outlined in Recording 5.1 to two simple probability questions. Since a lower-tail probability of a negative z value is required in the first case, recall the use of Φ (–k) = 1 - Φ(k) for some constant k. In the second part an upper-tail probability is obtained by noting that Pr(Z > z) = 1 - Pr(Z ≤ z) = 1 - Φ(z). Duration: 14 minutes 56 seconds.

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Video tutorial 5.3 Sampling distribution of

The standardisation technique is now applied to computing probabilities associated with the sample mean random variable,

. It should be emphasised that during the standardisation we now divide by the standard error, σ/√n, and not the standard deviation, σ, as in tutorials 5.1 and 5.2. Duration: 13 minutes 36 seconds.

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Video tutorial 5.4 Central Limit Theorem review

Following your background reading, the Central Limit Theorem (CLT) may still seem a little confusing. This recording aims to consolidate your previous reading to clarify the importance of the CLT. Duration: 6 minutes 48 seconds.

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Video 3.1 Sample variance and standard deviation

Discussion of the various different, but equivalent, formulae for the sample variance is presented with revision of the summation operator,

. Application to simple data set performed and standard deviation derived. Duration: 10 minutes 17 seconds.

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Video 3.2 Stem-and-leaf diagram

Key components of a standard stem-and-leaf diagram are presented with advice on how to maximise the number of marks when answering such a question in an examination. Consideration is also given to commenting on the shape of the distribution revealed by a stem-and-leaf diagram with reference to outliers and skewness. Duration: 20 minutes 29 seconds.

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Video 3.3 Histogram

Construction of a histogram is discussed with a review of the key points looked for by examiners. Due to bar areas being proportional to the frequency of observations, designing histograms to have equal category widths is recommended. Duration: 20 minutes 29 seconds.

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Chapter 4 - Video Tutorials index

Video tutorial 4.1 Set theory notation applied to a system of components

We consider a machine which has two components in parallel, hence the system works if at least one component is working. Given probabilities of the components working of 0.7 and 0.8 for components one and two respectively, the axioms of probability are applied to compute the probability of the system working. Duration: 11 minutes 13 seconds.

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Video tutorial 4.2 Venn diagram

A Venn diagram is used to model the intersection of sets. For a class of 35 students, information about the number studying combinations of the disciplines of Economics, History and Politics is provided. The Venn diagram then makes it possible to compute the probability of a student studying at least one of these subjects. Duration: 8 minutes 19 seconds.

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Video tutorial 4.3 Conditional probability

A horse race example is used to illustrate the concept of conditional probability. Using the idea of the Total Probability Theorem, it is possible to compute the probabilities of conditional events of interest. Duration: 7 minutes 40 seconds.

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Video tutorial 4.4 Probability trees

Sometimes it can be conceptually easier to use probability tree diagrams to set out solutions to a problem. This recording makes use of such a technique to compute the probability of an audit error when two individuals are responsible for checking audits. Duration: 10 minutes 26 seconds.

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Video tutorial 4.5 Rolling of dice

This probability example considers constructing the entire sample space of possible outcomes when two dice are rolled. Using this approach, the probabilities of particular events of interest are computed. Duration: 6 minutes 51 seconds.

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Video tutorial 1.1 Use of the summation sign, Σ

The summation operator, Σ, is explained with emphasis on the index of summation. The operator is extended to computing a sample mean followed by a simple problem using two sets of summary statistics to compute a combined sample mean.

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