Based on the row percentages table in a above, it is clear that health-concious consumers tend to purchase more fat-free dairy products than full cream products. It is a measure of the level of influence of the treatment factor on the response measure. Any differences can be attributed to or explained by the influence of the treatment factor on the numeric response measure.
Conclusion: All population means are equal. Therefore conclude that there is at least one motor vehicle type with a different average fuel consumption to the rest. By inspection, it would appear that VW has an average fuel consumption that is significantly different higher, and hence least fuel efficient from Peugot and Ford. Note: The sample evidence must be more convincing i. Assumption 2 A normally distribution population for the response variable. Therefore conclude that there is no difference in the mean volume of sales across the 3 package designs.
All are likely to generate the same average sales. Therefore the cereal producer can choose any of the three package designs for their new muesli cereal. Therefore conclude that there is at least one bank that has a different mean service rating score to the other banks. By inspection, it would appear that Bank X has a significantly higher mean service rating score than the other two banks.
The three banks are perceived similarly by customers in terms of their service levels. Note: The reason for the change in conclusion between a and b is that the statistical evidence is only weak i. Therefore conclude that there is at least one shelf height that generates a different mean level of sales to the other shelves. By inspection, it would appear that shoulder and waist high shelves generate higher average sales of the drinking chocolate product than bottom or top shelves. Therefore conclude that there is at least machine that has a different mean processing time.
By inspection, machine C has a significantly longer mean processing time than either machines A and B. Machine C must not be considered for purchase. The population mean processing times between the two machines A and B are likely to be identical. Thus the company can purchase either machine A or machine B.
Therefore conclude that there is at least one sector with a different mean earnings yield relative to the other sectors. Therefore conclude that there is at least one advertising strategy that results in a different mean level of deodorant sales relative to the other strategies. On average, the Sophisticated and Trendy strategies appear to be equally effective the difference in sample means does not appear significant.
Recommendation: Either the Trendy or the Sophisticated strategy can be adopted. Therefore the population mean sales from each of the two strategies is likely to be identical. The two strategies are therefore equally effective and either can be adopted by the company. It appears the least effective. The remaining two strategies Sophisticated and Trendy are equally effective and therefore either can be adopted by the company to promote its new ladies deodorant.
These differences do appear to be significant. Therefore conclude that there is at least one sector with a different mean leverage ratio relative to the other sectors. By inspection, the banking sector has the lowest mean leverage ratio, while construction and manufacturing appear to have similarly high mean leverage ratios.
Recommendation The investor is advised to consider either the banking sector with the lowest mean leverage ratio or the technology sector with a marginally higher mean leverage ratio. This difference may not be statistically significant. The population mean leverage ratios between the Technology and the the Banking sector are therefore likely to be equal. Their mean leverage ratios are likely to be equal. Since both sectors offer an investor the same lower risk, either or both can be chosen for investment.
Therefore conclude that there is at least one training method with a different mean performance score relative to the other training methods. By inspection, the lecture and audio-visual are the least effective lower mean scores , while on-the-job and the role-play methods are more effective with higher mean scores.
Recommendation The training manager is advised to consider either on-the-job training or use role play methods The difference does not appear to be statistically significant. The population mean performance scores between the two training methods is likely to be the same.
Both are likely to produce similar high mean performance scores. In two-factor ANOVA, two categorical factors are used to explain possible differences between the observed sample means. It is constructed from the sample means of the various combinations of the different factor levels. Alternatively, its p-value 0. If Arts graduates are employed, they must be given intensive marketing training.
Hence all switch devices are likely to have the same average dropped call rate. Factor: Interaction effect Since F-stat 4. By contrast, average rubber wastage of radial tyres is lowest on TAM3 3. Correlation analysis measures the strength of the relationship between the two numeric variables used in the regression equation. There is no statistically significant relationship between x and y. The more the training received, the higher the output.
This is a high level of explained variation. Hence training input is very beneficial to worker output and the training programmes should be continued. As inventory turnover increases, earnings yields also increases. Thus the business analyst's view is supported by the strong sample evidence. Hence inventory turnover has been shown to have a significant direct effect on a company's earnings yield. Yes, the regression equation can by used with confidence to estimate earnings yield based on a company's level of inventory turnover.
Conclude that there is a strong positive association between inventory turnover and earnings yield. Conclude that there is a strong direct association between age of machines and the level of annual maintance costs in Rands. Alternatively, for every year older, the annual maintenance costs increase by R5.
The association between aptitude score and performance rating is not statistically significant. Therefore, the call centre manager should have low confidence in this estimated performance rating score. Conclude that the overall model is statistically significant. For B : Since t-stat For C : Since t-stat 1. For D : Since t-stat 1. Hence, do not reject H0 if Conclude the overall model is statistically significant.
Satisfaction : Since t-stat Commitment : Since t-stat Yes, organisational commitment does play a statistically signficant role in explaining employee absenteeism see c , d and e above. Speed : Since t-stat 3. Viscosity : Since t-stat 1. MB : Since —t-crit Marketing methods B and C can be combined as there is no statistically significant difference between them with regards to their average productivity levels across consultant.
MA : Since its p -value 0. MB : Since its p -value 0. R2 Cape : Since t-stat 1. Example A quantity index measures changes in consumption levels over time, holding prices constant. A price relative is a change in the level of activity of an item in a given period relative to a base period.
This removes the influence of price increases. SmartAccess on the other hand showed an increase in price from to of However, the number of Programmers, on the other hand, reduced by The price of the HQ32 printer cartridge has decreased by 4.
However, the average price of print cartridges decreased by 0. In , prices were 2. Baltic Insurance showed a 6. Hence Federal Insurance showed the bigger claims increase from to Hence Baltic Insurance. The largerst price change increase was sugar with a Quantity c Food items Relatives Milk litres 98 The largest consumption change decrease was sugar with a It is interesting to note that sugar showed the largest unit price increase while simultaneously recorded the largest decrease in consumption from to Electricity showed the smallest change increase of 6.
A slight increase of only 0. From to , however, the average cost of household utilities actually decreased marginally by 2. Only sewage showed a decline in usage by 5. Relative to unit costs, the cost of leather goods inputs was higher by 8. Since , real salaries have declined relative to base and are continuing to fall further behind inflation CPI.
What is the sampling unit in this scenario? Why is it important that the sample of 68 HR managers be randomly selected? Does the sample evidence support their claim? What is the random variable of interest? What percentage of readers interviewed read Fair Lady regularly?
Is this a statistic or a parameter? Does the problem scenario require inferential statistics or only descriptive. Over the past six months they had varied both the number of ads placed per week and the advertising media press, pamphlets, magazines used each week. Weekly sales volume data was recorded, as well as the number of ads placed per week and the advertising media used each week. How many random variables are there in this study?
Name them. Which random variable is being predicted? Which random variables are assumed to be related to the variable being predicted? Which area of statistical analysis is suggested by this management scenario?
Scenario 1 South Coast Estate Agency wants to determine the average selling price per square metre and size of accommodation of all residential properties in Margate, KwaZulu- Natal. The data from the 25 residential properties sold by their agents, out of the total sales in the area last year, was gathered from deeds of sale documents.
Scenario 2 The owner of the Numbi Restaurant asked a sample of 18 patrons who ate at the restaurant on a particular Saturday evening to complete a short questionnaire to determine their perception of the quality of service and food received that evening. Scenario 3 The organisers of the Design for Living exhibition held annually at the Good Hope Centre, Cape Town conducted a survey during the latest exhibition by randomly selecting visitors as they left the exhibition hall.
Scenario 5 Metrorail, the train commuter service in Cape Town, has been working on improving service to its commuters. A random sample of commuters was interviewed recently on trains over a period of a week and asked their opinion on issues of personal safety on trains, comfort, cleanliness, convenience and punctuality.
The results of the sample are to be used to measure the improvement in service. Scenario 6 Metrorail also recently conducted a campaign to attract road bus, taxi and car commuters to using their rail service. The brief of the researchers was to estimate the percentage of road commuters that converted to train commuting as a result of the campaign. Scenario 7 The Star newspaper in Gauteng conducted a survey amongst a random cross-section of its subscriber readers to identify the popularity of the various sections of the newspaper amongst all its readers.
Also give two illustrative data values for each of these random variables:. Rank your preference of the fruit juices that you have just tasted. Orange . Guava . Apple . Grape . Do you enjoy your job? Which mode of transport do you mostly use to commute to work?
Car . Bus . Train . Taxi . Motorcycle . Bicycle . Rate your response to this statement using the Likert rating scale:. Strongly disagree  Disagree  Unsure  Agree  Strongly agree . How would you rate the service level of your bank? Use the following semantic differential rating scale:. Extremely poor . Very poor . Poor . Unsure . Good . Very good . Excellent . Financial analysis study.
Economic sector. Other specify. Head office region:. Western Cape. Free State. Company size in terms of number of employees. Turnover rand per annum :. More than 50 million. Share price in cents as at 31 December Earnings per share in cents for tax year. Dividends per share in cents for tax year.
Number of shareholders as at 31 December Return on investment for tax year. The data captured in this schedule is extracted from financial reports of JSE-listed companies and is used to compile a database on the financial status of JSE-listed companies.
How many random variables are being studied in the questionnaire? For each question, identify:. Give an illustrative data value for each random variable in the study. Voyager is an SAA customer loyalty programme. Voyager service quality questionnaire. Section B:. Voyager usage level. Did you encounter any problems in claiming Voyager Awards? Yes u. Car rentals. Hotels and resorts. Financial services. Section C:.
Voyager service quality perceptions. For each statement, indicate your level of support by circling the appropriate number. I receive my statements regularly. The Voyager Guide is user friendly. Voyager Centres are conveniently situated. Voyager staff have good communication skills. My queries are always dealt with effectively. Voyager staff are knowledgeable about their product. The Voyager Guide. Voyager partnership plan. Voyager in-flight services.
Voyager holiday specials. Thank you for completing this questionnaire. Summarising Data:. Summary Tables and Graphs. Managers can easily understand sample data when it is summarised into an appropriate table and then displayed graphically. This chapter explains how to summarise data into table format and then how to display the results in an appropriate graph or chart.
Managers can only benefit from statistical findings if the information can easily be interpreted and effectively communicated to them. Summary tables and graphs are commonly used to convey statistical results. A table or a graph can convey information much more quickly and vividly than a written report. In practice, an analyst should always consider using summary tables and graphical displays ahead of written texts, in order to convey statistical information to managers.
Summary tables and graphs can be used to summarise or profile a single random variable e. The choice of a summary table and graphic technique depends on the data type being analysed i. The sample dataset in Table 2. See Excel file C2. Table 2. Single Categorical Variable. Categorical Frequency Table. A categorical frequency table summarises data for a single categorical variable. It shows how many times each category appears in a sample of data and measures the relatively importance of the different categories.
Follow these steps to construct a categorical frequency table:. List all the categories of the variable in the first column. Count and record in the second column the number of occurrences of each category. Convert the counts per category in the third column into percentages of the total sample size. This produces a percentage categorical frequency table. It is always a good idea to express the counts as percentages because this makes them easy to understand and interpret.
In addition, it makes the comparisons between samples of different sizes easier to explain. A categorical frequency table can be displayed graphically either as a bar chart or a pie chart. Bar Chart. To construct a bar chart, draw a horizontal axis x -axis to represent the categories and a vertical axis y -axis scaled to show either the frequency counts or the percentages of each category. Then construct vertical bars for each category to the height of its frequency count or percentage on the y -axis.
However, neither the order of the categories on the x -axis, nor the widths of the bars matter. It is only the bar heights that convey the information of category importance. Pie Chart. To construct a pie chart, divide a circle into category segments. The size of each segment must be proportional to the count or percentage of its category. Example 2. A market research company conducted a survey amongst grocery shoppers to identify their demographic profile and shopping patterns.
A random sample of 30 grocery shoppers was asked to complete a questionnaire that identified:. The response data to each question is recorded in Table 2.
Each column shows the 30 responses to each question and each row shows the responses of a single grocery shopper to all six questions. Management Questions. To construct the percentage frequency table, first count the number of shoppers that prefer each store — there are 10 ones Checkers , 17 twos Pick n Pay and 3 threes Spar. Then convert the counts into percentages by dividing the count per store by 30 the sample size and multiplying the result by i.
The percentage frequency table of grocery store preferences is shown in Table 2. Preferred store. The relative importance of each category of the frequency table is represented by a bar in a bar chart see Figure 2. Charts and graphs must always be clearly and adequately labelled with headings, axis titles and legends to make them easy to read and to avoid any misrepresentation of information. The data source must, where possible, also be identified to allow a user to assess the credibility and validity of the summarised findings.
Bar charts and pie charts display the same information graphically. In a bar chart , the importance of a category is shown by the height of a bar , while in a pie chart this importance is shown by the size of each segment or slice. The differences between the categories are clearer in a bar chart , while a pie chart conveys more of a sense of the whole.
A limitation of both the bar chart and the pie chart is that each displays the summarised information on only one variable at a time. Two Categorical Variables. Cross-tabulation Table. A cross-tabulation table also called a contingency table summarises the joint responses of two categorical variables. This summary table is used to examine the association between two categorical measures.
Follow these steps to construct a cross-tabulation table:. Assign each pair of data values from the two variables to an appropriate category— combination cell in the table by placing a tick in the relevant cell. When each pair of data values has been assigned to a cell in the table, count the number of ticks per cell to derive the joint frequency count for each cell. Sum each row to give row totals per category of the row variable. Sum each column to give column totals per category of the column variable.
Sum the column totals or row totals to give the grand total sample size. These joint frequency counts can be converted to percentages for easier interpretation. The percentages could be expressed in terms of the total sample size percent of total , or of row subtotals percent of rows or of column subtotals percent of columns.
The cross-tabulation table can be displayed graphically either as a stacked bar chart also called a component bar chart or a multiple bar chart. Stacked Bar Chart. Follow these steps to construct a stacked bar chart:. Choose, say, the row variable, and plot the frequency of each category of this variable as a simple bar chart.
Split the height of each bar in proportion to the frequency count of the categories of the column variable. This produces a simple bar chart of the row variable with each bar split proportionately into the categories of the column variable. Note: The stacked bar chart can also be constructed by choosing the column variable first and then splitting the bars of the column variable into the category frequencies of the row variable.