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Business Statistics
In a delayed food shipment, \(p=40 \%\) have passed their expiry date. If you pick 14 random food items, what is the probability that 3,4 , or 5 have passed their expiry date?
The stochastic variable \(Z\) is the number of \(T\) from 20 trials in a Bernoulli process with parameter \(p=0.8\). Find the distribution function \(h\) of \(Z\), and calculate its expected value
The craftsmen where you live on average keep \(80 \%\) of their deadlines. You hire 20 craftsmen for an expansion of your home. What is the probability that no more than 5 go past their deadlines?
Assume that precisely \(30 \%\) of the population will be voting Liberal Democrat at the next election. You are conducting a poll. What is the probability that between 27 and \(33 \%\) of a random
DiscCo's latest 8TB hard drive has an error rate given by the fact that the probability of error in a given bit within a given hour is \(p=50 \times 10^{-15}\). (We use the fact that \(8
A repair protein by the name of helicase moves along a strand of DNA in a random walk driven by water molecules. If we assume an upwards movement with probability \(p=0.51\), and then down with
A stochastic variable \(X\) follows a hypergeometric distribution with parameters \(N=20, S=13\), and \(n=5\). Find the distribution function \(f\) of \(X\), and calculate its expected value and
In a bag of assorted candies, there are 13 Almond Joys and 7 Twizzlers left. You sample 5 candies at random. What is the probability that you get 2 or fewer Almond Joys?
A stochastic variable \(X\) follows a hypergeometric distribution with parameters \(N=30, S=7\), and \(n=9\). Find the distribution function \(f\) of \(X\), and calculate its expected value and
A bag contains 23 copper rings and 7 gold rings. You reach into the bag and grab 9 rings. What is the probability that at most 1 of them is a gold ring?
You have a hand in 5-card poker with 2 kings, 1 ace, and 2 numbered cards. You consider trading in the two numbered cards and the ace for three new cards. What is the probability that at least one of
X is Poisson distributed and \(\lambda=2.8\). Calculate \(\mu_{X}, \sigma_{X}, P(X=1)\), and \(P(X eq 1)\).
X is Poisson distributed and \(\mu_{X}=3.5\). Calculate \(\sigma_{X}, P(X=5)\), and \(P(X>0)\).
X is Poisson distributed and \(\sigma_{X}=5\). Calculate \(\mu_{X}, P(X=7)\), and \(P(X \leq 2)\).
X is Poisson distributed and \(P(X=0)=0.1\). Calculate \(\lambda, \mu_{X}, \sigma_{X}\), and \(P(X=1)\).
The stochastic variable \(X\) is Poisson distributed with parameter \(\lambda=2.37\). Find the distribution function \(f\) for \(X\), and calculate the expected value and standard deviation. What is
You have tested a lump of Cryptonite with a Geiger counter, and found that it gives an average of 2.37 clicks on the counter per minute. Find the probability of 1 click or less in the next minute.
The stochastic variable \(X\) is Poisson distributed with parameter \(\lambda=1\). Find the distribution function \(f\) of \(X\), and calculate the expected value and standard deviation. What is
The gambling joint Backstab Saloon by Crater Lake in Oregon evict an average of one cheat per hour. You are on your way to Seattle to talk to "Bill" about a lucrative job, but have an hour to spare
X, the number of foxes the Easter Bunny sees in the forest on a given day, is Poisson distributed. During the seven days of Week 5, Mr Bunny saw respectively \(1,5,4,7,3,2\), and 8 foxes. By an odd
The difference between the binomial and uniform distributions.(a) You toss a fair \(D_{12}\) whose faces have values \(1, \ldots, 12\). What is the probability of an outcome of 3 or less?(b) You toss
The PirateBay problem: Find the distribution for yourself! Is it uniform, binomial, hypergeometric, or Poisson? For some of the problems, you should use an approximation rather than try to calculate
There is a new toll ring around the town Autopol. There are some initial hiccups, so the company running the toll booths only get to check on one in every four cars passing through the toll gates.
Which (if any) of the following statements about P‐values is CORRECT?A. The P‐value from a hypothesis test is the probability of obtaining your results, or more extreme results.B. The P‐value
Which (if any) of the following statements about the Type I error is CORRECT?A. The Type I error is the probability of rejecting the null hypothesis when it is true.B. The Type I error is the
Which (if any) of the following statements about hypothesis testing is CORRECT?A. With a large sample size you will always calculate a large P‐value from your hypothesis test.B. The P‐value from
Table 6.4 reports that the 95% CI for the comparison of mean corn size at nine months between the corn plaster and scalpel groups is: −1.2 to 0.1 mm. The research literature suggests that a
Table 6.4 reports that the 95% CI for the comparison of mean corn size at 12 months between the corn plaster and scalpel groups is: −1.7 to −0.2 mm. The research literature suggests that a
Table 6.4 reports that the 95% CI for the comparison of mean corn size at three months between the corn plaster and scalpel groups is: −1.5 to−0.5 mm. The research literature suggests that a
For the data in Table 7.15, suppose we wish to compare the difference in EORTC QLQ‐C30 physical function domain scores before and after exercise. Which ONE of the following statistical methods
For the data in Table 7.15, the mean difference in EORTC QLQC30 physical function domain scores before and after exercise is:A. −9.0 B. −1.0 C. 16.3 D. 61.7 E. 78.0Table 7.15 EORTC QLQ-C30
For the data in Table 7.15, the standard error of the mean difference in EORTC QLQ‐C30 physical function domain scores before and after exercise is:A. 3.9 B. 4.8 C. 7.1 D. 9.0 E. 25.4Table 7.15
What are the degrees of freedom (df) for the t‐test for comparing the mean difference in EORTC QLQ‐C30 physical function domain scores before and after exercise for the data in Table 7.15?A. 6 B.
Calculate the value of the test statistic for the t‐test for comparing the mean difference in EORTC QLQ‐C30 physical function domain scores before and after exercise for the data in Table 7.14.A.
The mean EQ‐5D VAS outcomes in Table 7.16 were compared between the BA and control groups using a two independent samples t‐test.Which of the following IS NOT an assumption for the two
From the data in Table 8.11 what is the proportion of participants in the corn plaster group who had a completely healed corn at six‐month post‐randomisation?A. 0.30 B. 0.36 C. 0.64 D. 0.70 E.
From the data in Table 8.11 what is the proportion of participants in the scalpel group who had a completely healed index corn at six‐month post‐randomisation?A. 0.30 B. 0.37 C. 0.64 D. 0.70 E.
From the data in Table 8.11, what is the difference in response (proportion of participants who had a completely healed index corn at six‐month post‐randomisation) between the corn plaster and
From the data in Table 8.11 calculate a 95% confidence interval for the difference in response (proportion of participants who had a completely healed index corn at sixmonths post‐randomisation)
Which of the following statements IS NOT an assumption of the Pearson chi‐squared test for a contingency table in order for it to be valid?A. The data comprise two unordered categorical
For the completely healed index corn at six‐month postrandomisation outcome and data in Table 8.11 calculate the chisquared test statistic (without continuity correction) to compare the proportions
For the completely healed index corn at six‐month postrandomisation outcome and data in Table 8.11 calculate the degrees of freedom for the chi‐squared test statistic.A. 1 B. 2 C. 3 D. 4 E.
The P‐value from the chi‐squared test for comparing the completely healed index corn at six‐months post‐randomisation outcome and data in Table 8.11 is?A. 0.20Table 8.11 Randomised (n)
From the data in Table 8.11 the 95% confidence interval for the difference in proportions with a completely healed index corn at 12‐months post‐randomisation between the corn plaster and scalpel
Which of the following statements about the use and interpretation of correlation coefficients is FALSE?A. A correlation coefficient always lies in the range –1 to 1.B. A correlation coefficient
What do you think the correlation between X and Y is in the graph below?A. −1.0 B. −0.8 C. 0.0 D. 0.5 E. 1.0 Y
What do you think the correlation between X and Y is in the graph belowA. −1.0 B. −0.5 C. 0.0 D. 0.7 E. 1.0 Y X
Figure 9.11 shows a scatterplot and regression line of the relationship between the between distance walked, in metres on an endurance shuttle walk test before (pre) and after (post) a physiotherapy
From Figure 9.11 what do you think is the approximate intercept of the regression line for the relationship between pre and post rehabilitation distance walked?A. 0 B. 100 C. 200 D. 300 E. 400Figure
From Figure 9.11, what do you think is the approximate slope of the regression line for the relationship between pre and post rehabilitation distance walked?A. 0 B. −1.2 C. −10 D. 1.2 E.
Using the regression line, in Figure 9.11, what is the predicted post‐rehabilitation distance walked given a pre‐rehabilitation distance walked of 300 m?A. 0 B. 100 C. 200 D. 550 E. 800Figure
Table 9.10 below shows the estimated regression coefficients from the multiple linear regression model to predict post‐rehabilitation distance walked, in metres, on an endurance shuttle walk test
From Table 9.10 and model what is the predicted post‐rehabilitation distanced walked in metres, for a man, aged 60, with a pre‐rehabilitation distance walked of 100 m?A. 4.5 m B. 92.3 m C. 337.2
From the Table 9.10 and the model what is the proportion of the variability in the outcome, post‐rehabilitation distance walked, which is NOT explained by age, gender and pre‐treatment distance
Calculate the missing Wald test statistics value for cell WWWW (the randomised treatment group variable/row).A. 3.298 B. 1.657 C. 0.822 D. −0.298 E. −3.455
Which variables ARE statistically significantly associated with the binary outcome recurrent haemorrhoids at 12 months after the procedure?A. Age only B. Sex only C. Previous treatment only D.
Calculate the missing odds ratio value for cell YYYY (the randomised treatment group variable/row). The odds ratio for the randomised treatment group variable is:A. 0.45 B. 0.88 C. 1.00 D. 1.21 E.
Calculate the missing upper and lower limits for the 95% CI estimate for the Odds Ratio (the randomised treatment group variable/row) for cells ZZ and ZZZZ. The 95% CI for the odds ratio for the
Table 10.10 reports that the 95% CI for the odds ratio of a recurrent haemorrhoid at 12 months between males and females, after adjustment for age, previous treatment and group is:0.77 to 1.90.Which
Calculate the predicted probability of recurrence of a haemorrhoid at 12 months for a female patient, aged 50, with no previous treatment in the RBL group.A. 0.25 B. 0.29 C. 0.34 D. 0.39 E. 0.43
Calculate the predicted probability of recurrence of a haemorrhoid at 12 months for a female patient, aged 50, with no previous treatment in the HAL group.A. 0.25 B. 0.29 C. 0.34 D. 0.39 E. 0.43
The sample size calculation for the trial was based on detecting a 15% absolute difference in the proportion of patients who experience a recurrence of a haemorrhoid at 12 months between the RBL and
Assume the odds ratio calculated from question 10.10 would be regarded as the smallest effect that was clinically or practically important. Then, using the 95% CI, calculated in 10.6, the odds ratio
From Figure 11.9 what is the approximate median survival time, in days, from gastrostomy to death in the PIG group?A. 100 B. 200 C. 300 D. 325 E. 375Figure 11.9 Overall survival 1 0.75 0.50 0.25 0
From Figure 11.9 what is the estimated median survival time, in days, from gastrostomy to death in the RIG group?A. 100 B. 200 C. 300 D. 325 E. 375Figure 11.9 Overall survival 1 0.75 0.50 0.25 0
From Figure 11.9 how many patients (in total) with amyotrophic lateral sclerosis were at risk of dying at 300 days?A. 319 B. 248 C. 185 D. 125 E. 0Figure 11.9 Overall survival 1 0.75 0.50 0.25 0
From Figure 11.9 what are the degrees of freedom for the logrank test?A. 1 B. 2 C. 3 D. 125 E. 318Figure 11.9 Overall survival 1 0.75 0.50 0.25 0 Number at risk 0 Log-rank test = 11.40 on X df, P =
From Figure 11.9 the P‐value from the Logrank test is P = 0.003.Which of the following statements are true?A. This result is not statistically significant.B. The probability of getting the value of
From Table 11.4, what type of variable is ‘site of symptom onset’?A. Discrete B. Nominal C. Binary D. Ordinal E. ContinuousTable 11.4 Variable Gastrostomy insertion method B SE(B) z P-value Ratio
Using the data in Table 11.4 calculate the missing test statistics value for cell WWWW (the PIG vs PEG variable/row).A. 1.043 B. 1.657 C. 0.822 D. −0.298 E. −3.458Table 11.4 Variable Gastrostomy
From Table 11.4 which variables ARE statistically significantly associated with the hazard or risk of dying after the gastrostomy procedure?A. Age only B. Weight only C. Site of symptom onset D.
Using the data in Table 11.4 calculate the missing hazard ratio value for cell YYYY (the PIG vs PEG variable/row).A. 0.45 B. 0.88 C. 1.00 D. 1.41 E. 2.20Table 11.4 Variable Gastrostomy insertion
Using the data in Table 11.4 calculate the missing upper and lower limits for the 95% confidence interval estimate for the Hazard Ratio (the PIG vs PEG variable/row) for cells ZZ and ZZZZ.A. 0.58 to
From the data in Table 12.7 calculate the estimated probability that the two radiographers will agree on the diagnosis of a vertebral fracture.A. 0.09 B. 0.14 C. 0.50 D. 0.72 E. 0.86Table 12.7 Any
From the data in Table 12.7 calculate the Kappa statistic.A. 0.09 B. 0.14 C. 0.50 D. 0.72 E. 0.86Table 12.7 Any fracture from DXA scan Observer 2 Total Observer 1 No Yes Total No Yes 23 6 29 1 20 21
The Kappa value from 12.2 indicates which of the following strengths of agreement:A. Poor B. Slight C. Moderate D. Good E. Very High
Calculate Cronbach's alpha for the data in Table 12.8.A. 0.09 B. 0.14 C. 0.50 D. 0.72 E. 0.86Table 12.8.Q1 Does someone give you extra help to keep you safe?Q2 Do you keep away from busy areas to
Which of the following are TRUE?A. If items of a scale are all essentially measuring the same thing then Cronbach's α is expected to be close to 1.B. Cronbach's α can be used with nominal
Which of the following are TRUE?A. A measurement with a CV of 50% is repeatable B. The Bland–Altman method can be used to compare instruments with different scales.C. The limits of agreement will
12.7 Calculate the ICC for the data in Table 12.9.A. 0.09 B. 0.14 C. 0.37 D. 0.72 E. 0.86Table 12.9 CLD (mm) Subject First occasion Second Difference Mean occasion 1 18 15 -3 16.5 2 14 13 -1 13.5 3
What is the prevalence of disease (bowel cancer) in this sample?A. 5%B. 10%C. 15%D. 20%E. 25%
What is the sensitivity of the new test?A. 80%B. 89%C. 94%D. 95%E. 98%
What is the specificity of the new test?A. 80%B. 89%C. 94%D. 95%E. 98%
What is the PPV of the new test?A. 80%B. 89%C. 94%D. 95%E. 98%
What is the NPV of the new test?A. 80%B. 89%C. 94%D. 95%E. 98%
What proportion of individuals in the sample were correctly diagnosed by the new test?A. 80%B. 89%C. 94%D. 95%E. 98%
Which ONE of the following statements about the sensitivity of a diagnostic test is INCORRECT?A. Given the patient has the disease, sensitivity is the proportion of times the test is positive.B.
Which ONE of the following statements about the specificity of a diagnostic test is INCORRECT?A. Given the subject is free of disease, specificity is the proportion of times the test will be
Which ONE of the following statements about diagnostic tests is INCORRECT?A. The PPV is the proportion of patients with positive test results who are correctly diagnosed.B. The NPV is the proportion
Which ONE of the following statements about the ROC curve(when the results of a diagnostic test are cut‐points in a continuum) is INCORRECT?A. The ROC curve is obtained by plotting sensitivity vs
From the data in Table 14.11 what proportion of subjects in the exposed (current or ex‐smoker)group died from any cause in the 20‐year follow‐up?A. 0.162 B. 0.292 C. 0.319 D. 0.681 E.
From the data in Table 14.11 what is the risk of dying from any cause in the 20‐year follow‐up in the unexposed (non‐smoker) group?A. 0.162 B. 0.292 C. 0.319 D. 0.681 E. 0.838 Table 14.11 Died
From the data in Table 14.11 what is the absolute risk difference (difference in proportions) for dying from any cause in the 20‐year follow‐up between current or ex‐smokers and
From the data in Table 14.11 what is the relative risk for dying for current or ex‐smokers compared to non‐smokers?A. 0.41 B. 0.51 C. 1.97 D. 2.03 E. 2.43 Table 14.11 Died from any cause Current
Using the data in Table 14.11 calculate the 95% confidence interval for the relative risk for dying for current or ex‐smokers compared to non‐smokers?A. 0.61 to 0.74 B. 0.74 to 1.86 C. 1.17 to
Using the data in Table 14.12 what is the odds ratio for lung cancer cases compared to controls for being a smoker?A. 0.34 B. 1.05 C. 2.97 D. 11.02 E. 32.76Table 14.12 Smoking status Cases Controls
Using the data in Table 14.12 calculate the 95% confidence interval for the odds ratio for lung cancer cases compared to control for being a smoker?A. 0.64 to 1.54 B. 0.68 to 1.66 C. 1.79 to 4.95 D.
From the data on Table 14.13; what is the odds ratio for cases compared to controls for any abuse (verbal or emotional, physical, sexual) before age 19?A. 0.18 B. 0.45 C. 2.21 D. 3.23 E. 5.70Table
Using the data in Table 14.13 calculate the value of McNemar's test statistic without continuity correction for investigating the association between an ACE and bilateral oophorectomy.A. 0.18 B. 1.30
Using the data in Table 14.13 calculate the 95% confidence interval for the odds ratio for cases compared to controls for any abuse (verbal or emotional, physical, sexual) before age 19?A. 1.00 to
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