The correct answer is d). a) The null hypothesis is either true or false. b) The alternative hypothesis is either true or false. c) The P value is the probability of obtaining the observed or more extreme results if the null hypothesis is true. e) The cut-off of 0.05 is usually used to indicate significance, but the P-value can take any value between 0 and 1.

 The correct answer is d). a) The null hypothesis assumes no effect. Therefore, the null hypothesis could be “The difference in alcohol consumed by men and women is zero.” b) Normally we do not specify a direction for the difference. We allow for the difference to be greater or less than zero. This is known as a two tailed test. c) The P value is the probability of obtaining our results, or something more extreme, if the null hypothesis is true. The null hypothesis is either true or false. d) Typically we set the significance value for a test at 0.05. If we obtain a P value of less than or equal to 0.05 then we would reject the null hypothesis and accept the alternative hypothesis.

 The correct answer is b). a) A suitable null hypothesis is that subjects from families with genetic disorders in the population are equally likely to experience problems when applying for life insurance as those from families in the general population. c) We do not specify a direction for the alternative hypothesis except in very rare circumstances. Therefore, a suitable alternative hypothesis is that subjects from families with genetic disorders in the population are not equally likely to experience problems when applying for life insurance as those from families in the general population. e) As the P-value is so small, we believe that these results are unlikely to have arisen by chance.

 The correct answers are a) and b). c) We can reject the null hypothesis at the 5% level of significance (indeed we can reject it at the 0.1% level of significance). d) There is a less than 1 in 1000 chance that we would have obtained these results, or more extreme results, if the null hypothesis was true. e) We do not know that the workplace bullying has caused these results, only that individuals who have been bullied have lower levels of job satisfaction and higher levels of stress than those who have not been bullied. It may be that these individuals already had lower levels of job satisfaction and higher levels of stress before being bullied.

 The answers are c) and d). c) A Type II error represents a false negative. One way to remember this is that II is similar to N, a \ has been added to II get an N, whereas a Type I error is similar to P, a ⊃ has been added to I to get P. d) A type II error (β) is typically set at 0.1 or 0.2 (occasionally quoted as a percentage 10% or 20%). Power is typically quoted as 80% or 90% and is 1 - β.

 The answer is d). a) Probabilities for Type II error are called α (alpha). b) Type I error is often set at 5% or 0.05 but sometimes we use other values, for example 1% or 0.01. This corresponds to 95% confidence. c) A Type I error represents a false positive. So we conclude a significant effect even though no true effect exists. One way to remember this is that I is similar to P, a ⊃ has been added to I get a P, whereas a Type II error is similar to N, a \ has been added to II to get N.

 The answer is b). This is an example of a False Negative which is a Type II error.

 The answer is c). 1- β (beta) represents the Power of the study. It is 1 minus the probability of the Type II error (β) which is usually set at 0.02 or 0.01 (20% or 10%). Thus Power is typically 80% or 90%.

 The answers are b) and c). a) If the standard deviation is increased it means the amount of variation (size of error) is greater. We want the minimum amount of variation so the smaller the standard deviation the better. Remember the variance is the standard deviation squared. b) Increasing the size of the sample will reduce the amount of variation and improve the Power of the study. c) If you increase the probability of making one type of error (Type I or II), the probability of making the other type of error (Type I or II) decreases, assuming everything else stays the same. So if alpha is increased from 0.05 to 0.1, the probability of making a Type I error increases, an consequently the probability of making a Type II error decreases. d) If you increase the size of β (beta) from 0.10 to 0.20 you are increasing the chance of a False Negative and reducing the Power of the Study which is 1- β*100%. In an ideal world you choose have 100% power so β would be 0. If you would like to explore the topic further, you can use the CurveApplet from the Amazing Applications of Probability and Statistics by Tom Rogers.

 The correct answer is a). The estimated range is calculated from a sample of the population data. b) is false. The interval is a measure of precision of the estimate of the population parameter. The normal range gives information about individual values. c) is false. The greater the confidence you require, the wider the confidence interval will be. d) The more observations you include in your sample the smaller the standard error will become and a confidence interval is the estimate ± level of confidence factor * the standard error of the estimate. The ‘level of confidence factor’ depends on the level of confidence required, 90% = 1.68, 95% = 1.96 AND 99% = 2.58.

 The correct answers are b), c) and e). a) A confidence interval can be calculated for a small sample using the exact Binomial distribution. It will be very wide if the sample size is small. c) This statement is not strictly true, but the confidence interval is usually interpreted in this way. d) The 95% confidence interval is calculated as the sample proportion ± 1.96 times the standard error of the proportion. The 1.96 is often approximated by 2. The sample proportion ± standard error of the proportion is the 67% confidence interval for the proportion. e) A 95% CI is narrower that a 99% CI. We are accepting less precision for our estimated value because we want to be more certain to include the population estimate in the interval with our 99% interval.

 The correct answer is b). The 95% confidence interval is calculated as the sample mean ± 1.96 times the standard error of the mean. The standard error is calculated as the standard deviation divided by the square root of the number of observations in the sample. The 1.96 is often approximated by 2. In our example we would use: 90 ± 2 * 10 / 25 = 90 ± 20 /5 = 90 ± 4, which is 90 – 4 to 90 + 4 or 86 to 94.

 The correct answer is c). a) We are trying to estimate the weight loss in population. We know the weight loss in the sample. Our standardised consultation group weight loss is 0.74 Kg greater (1.11-0.37) than in the usual care group. The confidence interval does not describe a range of values for which 95% of the sample members achieved a weight loss. b) The confidence interval does not describe the range of values for individuals within population. The range that 95% of individuals could take is the reference range. c) We are estimating the population parameter using a sample. The population estimate has error associated with it and this is represented by the confidence interval. d) We know the sample weight loss in the standardised group exactly it is 1.11. There is no uncertainty about it.

 The answer is c). It is best to quote the confidence intervals to the same number of decimal places as the estimated parameter, the mean difference, and always use the word ‘to’ to distinguish between the lower and upper limits. Always quote the level of confidence, in this example 95%.

 The correct answer is d). P values should be quoted to two decimal places and not replaced with ns or a greater than 0.05. In this example it was not significant at the 5% level, but only just, so this is useful information for the reader.

 The correct answers are a) and b). a) If the confidence interval for a difference does not include zero then the difference will be statistically significant. b) If the confidence interval for a difference does not include zero then the difference will be statistically significant even it is a negative range. c) For risks and ratios, 1 is the value of no difference. A confidence interval that includes 1 indicates a lack of statistical significance for a risk or a ratio. d) This is a bit of a trick question. You cannot obtain a negative ratio. A confidence interval for a risk or a ratio will never cross zero, the lower limit value will be above zero.

 The correct answer is c). a) Generally, when 95% confidence intervals for two sample means or proportions overlap, inferences cannot be made about the presence or absence of statistical significance at the 5% level. In this example the difference in the cholesterol reduction between drugs A and B was 0.10 mmol/L and the 95% confidence for the difference was (0.03 to 0.18) which does include zero, so the difference will be significant at the 5% level. b) The two confidence intervals overlap.

 The correct answer is d). This is an extension of the simple bar chart, but instead of displaying the frequency counts or percentages of one categorical variable, it displays two; one categorical variable is nested within another. In a clustered bar chart, the horizontal axis represents the levels or categories of the nested variable, the bars within each cluster represent the categories of the other variable, while the vertical axis represents the frequency counts or percentage. The special feature of this graph is that there is no gap between the bars within a cluster, but there is gap between bars across the cluster. It is often clearer to use different colours on the bars, one colour for each clustered category.

 The correct answer is b). A box and whisker plot (or box plot) illustrates the location and spread of a continuous variable, highlighting any extreme values. The central box, which contains the middle 50% of the observations, extends from the lower quartile, LQ, of the data to the upper quartile, UQ, with the median being marked by an internal line across the box. This divides the 50% of observations into two groups of 25%. The difference between the upper and lower quartiles is known as the inter-quartile range (IQR). Whiskers are drawn from each end of the box extending as far as 1.5*IQR, or as far as the furthest observation within that range if less. Any observations lying further out up to 3*IQR are known as outliers, and any observations lying even further than 3*IQR are known as extreme values; these are drawn as separate dots. Two box plots, each representing a level in a categorical variable, can be drawn side by side to allow comparison.

 The correct answer is c). A scatter plot is always used to display the association or the relationship between 2 continuous variables. Each of these 2 variables is represented on the X and Y axis accordingly. Each point is plotted based on the corresponding pair of values for the observation, i.e. coincident point. The trend of the relationship indicated by the points represents the direction of the relationship between the two variables. The tighter the coincident points are to this trend then the stronger the relationship between the two variables. The special feature of this graph is that a cloud of dots are floating between the two axes.

 The correct answer is a). A line plot is usually used when there is a time variable involved. It is used to demonstrate a trend of a continuous variable throughout time. The Y axis represents the continuous variable, while the X axis represents the time variable. The chart is produced by modifying a scatter plot so that the points are joined together by lines with respect to the chronological order of the time variable. Some software packages offer a Line Chart but this is different, the X axis in a line chart is treated as categorical and not a true continuous or scaled axis. If the interval between items on the X axis on a line chart are equal (e.g., 0,1,2,3,4,5,6), the plot will look correct. If they are not equally spaced (0,1,3,6,12), the plot will be incorrect.

 The correct answer is a). The relationship between the variables Age and Height can be illustrated by scatter plot, since both variables are continuous and each axis can be used to represent one variable. Each point on the plot represents the observation with the corresponding value from the two variables.

 The correct answer is c). The relationship between the variables Height and Gender can be illustrated by a box and whisker plot, since each of the boxes represents the distribution of Height for one of the Gender groups.

 The correct answer is d). The relationship between the variables Smoke and Gender can be illustrated by cluster bar charts, since both of the variables are categorical. This chart displays the number (or proportion) of observations in each of the categories.

 The correct answer is b). A scatter plot should be used to illustrate the relationship between two continuous variables.

 The correct answer is c). This statement is describing the strength and direction of the relationship between the two continuous variables (BMI and age). A scatter plot is the best graph to use since it displays the association between 2 quantitative, generally continuous, variables. Each of these 2 variables represents the X and the Y axis accordingly. Each point is plotted based on the corresponding pair of values for the observation, i.e. coincident point. The trend of the relationship indicated by the points represents the direction of the relationship between the two variables. The closer the points are to forming a line, then the stronger the relationship between the two variables.

 The correct answer is c). This statement is describing the proportion of subjects in each of the categories formed by the variables smoking status and gender. One categorical variable is nested within another (smoking status nested within gender). A cluster bar chart is the best graph to use since it can display the frequency counts or percentages of two categorical variables. It is the most effective way to compare frequency counts or percentages graphically between the categories. In a cluster bar chart, the horizontal axis represents the levels or categories of the nested variable (e.g. smoking status), the bars within each cluster represents the categories of the other variable (e.g. gender), while the vertical axis represents the frequency counts or percentage.

 The correct answer is a). This statement is describing the distribution of a continuous variable (pain score) broken down by a categorical variable (age group). A box and whisker plot is the best graph to use since it illustrates the median, the lower quartile and the upper quartile of the data. It also highlights any observations that are extreme values. The central box, which contains the middle 50% of the observations, extends from the lower quartile, LQ, of the data to the upper quartile, UQ, with the median being marked by an internal line across the box. This divides the 50% of observations into two groups of 25%. The difference between the upper and lower quartiles is known as the inter-quartile range (IQR). Whiskers are drawn from each end of the box extending as far as 1.5*IQR, or as far as the furthest observation within that range if less. Any observations lying further out up to 3*IQR are known as outliers, and any observations lying even further than 3*IQR are known as extreme values; these are drawn as separate dots. Two box plots, each representing a level in a categorical variable, can be put drawn side by side to allow comparison.

 The answer is 0.063. The Proportion of infections in Hospital A is calculated by dividing the number with an infection in the hospital by the total number of cases in the hospital. In this case it is 16/253=0.063

 The answer is 0.045. The Odds of infection in Hospital B is calculated by dividing the number with an infection in the hospital by the number without an infection in the hospital. In this case it is 27/594=0.045.

 The answer is 6.3%. The Percentage of infections in Hospital A is calculated by dividing the number with an infection in the hospital by the total number of cases in the hospital and then multiplying by 100. In this case it is (16/253)x100=6.3%.

 The answer is 0.020. The difference in proportions is the proportion in one Hospital minus the proportion in the other. We are asked to calculate Hospital A – Hospital B. The proportion in Hospital A is 16/253=0.063… and the proportion in Hospital B is 27/621=0.043… The difference in proportions is therefore 0.063…-0.043…=0.020 to 3 decimal places. Try to keep all of the precision in your working until you get to the final answer.

 The answer is 1.45. The Relative Risk is the proportion in one Hospital divided by the proportion in the other. We are asked to use Hospital B as the reference group, which means we need to divide by Hospital B. The proportion in Hospital A is 16/253=0.063… and the proportion in Hospital B is 27/621=0.043… The Relative Risk is therefore 0.063…/0.043…=1.45 to 2 decimal places. Try to keep all of the precision in your working until you get to the final answer.

 The answer is 1.49. The Odds ratio is the Odds in one Hospital divided by the Odds in the other. We are asked to use Hospital B as the reference group, which means we need to divide by Hospital B. The Odds in Hospital A are 16/237=0.0675… and the Odds in Hospital B are 27/594=0.045… The Odds ratio is therefore 0.0675…/0.045…=1.49 to 2 decimal places. Try to keep all of the precision in your working until you get to the final answer.

 The answer is 0.0030. The Proportion of early neonatal deaths in Male babies is calculated by dividing the number of male babies who died in the early neonatal period by the total number of male babies. In this case it is 104/34261=0.0030.

 The answer is 0.0023. The Odds of early neonatal deaths in Female babies are calculated by dividing the number of female babies who died in the early neonatal period by the number of female babies who survived past the early neonatal period. In this case it is 75/(32494-75)=75/32419=0.0023.

 The answer is 0.00073. The difference in proportions is the proportion of early neonatal deaths in one gender minus the proportion in the other. We are asked to calculate Male– Female. The proportion in Male babies is 104/34261=0.0030… and the proportion in Female babies is 75/32494=0.0023… The difference in proportions is therefore 0.0030-0.0023…=0.00073 to 5 decimal places. Try to keep all of the precision in your working until you get to the final answer.

 The answer is 1.32. The relative risk for an early neonatal death is the proportion of early neonatal deaths in one gender divided by the proportion in the other. We are asked to use Females as the reference group and so we need to divide by the Female result. The proportion in Male babies is 104/34261=0.0030… and the proportion in Female babies is 75/32494=0.0023… The relative risk is therefore 0.0030…/0.0023…=1.315 to 3 decimal places. Try to keep all of the precision in your working until you get to the final answer.

 The answer is 1.32. The Odds ratio for an early neonatal death is the Odds of early neonatal deaths in one gender divided by the Odds in the other. We are asked to use Females as the reference group and so we need to divide by the Female result. The Odds in Male babies are 104/(34261-104)=104/34157=0.0030… and the Odds in Female babies are 75/(32494-75)=75/32419=0.0023… The relative risk is therefore 0.0030…/0.0023…=1.32 to 2 decimal places. Try to keep all of the precision in your working until you get to the final answer.

 The answer is 0.901. The Proportion of infections in School C is calculated by dividing the number vaccinated in the school by the total number of children in the school. In this case it is 581/645=0.901

 The answer is 9.08. The Odds of vaccination in School C are calculated by dividing the number vaccinated in the school by the number not vaccinated in the school. In this case it is 581/64=9.08.

 The answer is 0.178. The Odds of not being vaccinated in School C are calculated by dividing the number not vaccinated in the school by the number vaccinated in the school. In this case it is 127/714=0.178.

 The answer is -0.081. The difference in proportions is the proportion in one School minus the proportion in the other. We are asked to calculate School A – School C. The proportion in School A is 1003/1233=0.813… and the proportion in School C is 581/645=0.901… The difference in proportions is therefore 0.813…-0.901…=-0.081 to 3 decimal places. Try to keep all of the precision in your working until you get to the final answer.

 The answer is 1.25. The Relative Risk for not being vaccinated is the proportion in one School divided by the proportion in the other. We are asked to use School B as the reference group, which means we need to divide by School B. The proportion not vaccinated in School A is 230/1233=0.186… and the proportion not vaccinated in School B is 127/841=0.151… The Relative Risk is therefore 0.186…/0.151…=1.24 to 2 decimal places. Try to keep all of the precision in your working until you get to the final answer.

 The answer is 0.480. The Odds ratio is the Odds in one School divided by the Odds in the other. We are asked to use School C as the reference group, which means we need to divide by School C. The Odds in School A are 1003/230=4.360… and the Odds in School C are 581/64=9.078… The Odds ratio is therefore 4.360…/9.078…=0.480 to 3 decimal places. Try to keep all of the precision in your working until you get to the final answer.