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STAT108 - Interactive Exercises Notes (5-11)
Statistics for Commerce (STATS108)
University of Auckland
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Remember! Main Point
Statistics Revision
To estimate the standard deviation (SD) of a graph: - Determining the range: higher difference between the smallest and largest values usually indicates a higher SD value. - Density of data points: Concentration around the mean suggests lower SD, even spread or multiple peaks indicates higher SD - As an estimate SD id approx ¼ of the Interquartile range, third quartile - first quartile.
Chapter 5 (Estimation)
Sampling Variation: The change in statistic calculated each time you take a sample from a population.
Confidence Interval: The range of plausible values for a parameter (to express uncertainty about the true value of an unknown parameter).
Estimate: Our starting point for estimating the value of parameters is to calculate the relevant statistics for the sample data.
Sample-to-population inference: Use of sampling data to form generalisations about the characteristics of a population.
Remember: Sample proportion is a small sample of a large proportion, while allows us to provide valuable information about the distribution of a population without going through a lot of data.
CONFIDENCE INTERVAL INTERPRETATION (BOOTSTRAP CI FROM VIT)
CI generated limited of 0 and 0 in a study to see the proportion of kiwi customers say that banks can be trusted
With approximately 95% confidence, it is estimated that the percentage of kiwi customer say that banks can be trusted is somewhere between 47% and 51%
If we use the confidence intervals constructed to evaluate the claim “fewer than half of kiwi customers say that banks can be trusted” we can say that the claim is not supported (is supported) by the confidence interval as it is plausible (not plausible) that the proportion of kiwi customers that say banks can be trusted is above 50%
A percentage of 50% is one of the plausible values provided by a VIT bootstrap confidence interval for the proportion of kiwi customers that say that banks can be trusted.
Graph Analysis Q~ What is the mean total time in seconds of all audiobooks on LibriVOx
- Population: All audiobooks from Librivox
- Sample: 44 audiobooks from LibriVox
- Parameter
- Mean total time of all audiobooks available from LibriVox
- Extimate: Mean total time of 44 audiobooks from LibriVox
Remember: Positively skewed = sample mean > sample median
The distribution of total time hours is positively skewed (negatively skewed), which is why the sample mean is higher than the sample median.
CI bootstrap generated limits of 4 and 8.
Remember: prediction interval is different from an interval estimate for the population mean, therefore do NOT say “can be predicted to be”.
With approximately 95% confidence, it is estimated that the mean total time of audiobooks on libriVox is somewhere between 4 and 8 hours
Comparison Plastic percentages represents the percentage of litter collected that was plastic
bootstrap confidence interval constructed using the sample of boat trailers will be wider (narrower) than one constructed using the sample of mopeds.
Chapter 6 (Quantification)
Point Estimate = (Lower limit + upper limit) / 2 Width of the confidence interval: upper limit - lower limit Margin of Error: Point Estimate - upper limit
Sample Proportion Q~ Do you play mobile phone games for less than 60 minutes per week?
Remember: Use Confidence Interval Calculator
Claim: in 2010 the majority of adults played mobile games for less than 60 minutes a week.
The proportion of of survey participants (n = 814 ) in 2010 who played mobile games for less than 60 minutes per week was 68% (554/814x100)
BOTH ABOVE 50%
The 95% confidence interval for the proportion of adults who played mobile games for less than 60 minutes a week is between 64% and 71%.
Because the entire interval is above 50%, we can conclude that the majority of adults did play mobile games for less than 60 minutes a week.
SOME OR NONE ABOVE 50%
Because some (none) of the interval is above 50%, we can not conclude that the majority of adults did play mobile games for less than 60 minutes a week.
Difference between two sample proportions
whether the proportion of adults who played mobile games for less than 60 minutes per week has decreased from 2009 to 2010.
Remember: Use the calculator, but do difference between two sample proportions
The sampling situation for this comparison of proportions is two independent group (one group)
The proportion of survey participants in 2009 who played mobile games for less than 60 minutes per week was 79% (521/659x100)
The proportion of of survey participants (n = 814 ) in 2010 who played mobile games for less than 60 minutes per week was 68% (554/814x100)
The difference between the two sampling proportions is 0 - 0 = 0.
ONLY POSITIVE OR NEGATIVE VALUES
With 95% confidence, the proportion of US adult mobile gamers playing less than 60 minutes in 2009 was somewhere between 6 and 15 percent points higher (lower) than the proportion of adult playing less than 60 minutes in 2010.
As the confidence interval contains only positive differences (only negative differences) we can claim that the proportion of mobile gamers playing games for less than 60 minutes a week decreased (increased) from 2009 to 2010.
BOTH POSITIVE AND NEGATIVE VALUES With 95% confidence, the proportion of US adult mobile gamers playing less than 60 minutes in 2009 was somewhere between 1 percentage points lower and 15 percent points higher than the proportion of adult playing less than 60 minutes in 2010.
As the confidence interval contains both positive and negative differences we can not claim that the proportion of mobile gamers playing games for less than 60 minutes a week decreased (increased) from 2009 to 2010.
Sample Mean Sample mean: 23 m Sample SD: 4.
When comparing delivery times, the mean delivery time is between 1 minutes and 8. minutes higher than the other pizza place (with 95% confidence). As the confidence interval contains only positive values, there is enough evidence to conclude that the other pizza deliveries are faster in general.
Chapter 7 (Confirmation)
P value: measures how compatible the observed value is with the null model.
Normal distribution Remember: - the highlighted outcomes in the SD x 2 and then subtracted and adds from both ends of the graph - Normal Distribution Explorer
Mean height: 172 cm SD: 35 cm
35 x 2 = 70 172 +/- 70 = highlighted outcomes
Remember: larger tail proportion = less unusual, as long as it’s inside the 95%
According to the graph, a height of 210 would be usual (unusual) for this distribution, as they would have a right (left) tail proportion of 0.
Null Hypothesis + Alternative Hypothesis Remember: Null hypothesis takes a sceptical view, there is nothing new or interesting happening. Always use a two-sided hypothesis if you're not sure of the alternative.
Ex. Null: The proportion of all flights that experience delays is the same for AirNZ and JetStar. Alternative: The proportion of all flights that experience delays is not the same for AirNZ and JetStar.
Null: The mean price of all domestic flights for AirNZ is the same as the mean price of all domestic flights for Jet Star. Alternative: The mean price of all domestic flights for AirNZ is either higher or lower than the mean price of all domestic flights for Jet Star.
The alternative hypothesis can be one sided or two sides (can only be higher or lower, or can be both, respectively)
HYPOTHESIS TEST A sample of adults were given one of two options: a safe bet - guaranteed win of $30 or a higher payoff - 80% chance of winning $45.
Remember: If a value is 2 sided, you can add the values of both sides together( time by 2) and that would be the p value.
There were 394 respondents in the study. The group sizes were reasonably balanced (very unbalanced)
The estimate for the difference in mean age_perceived between the two groups (older than 30 - 30 or younger) is 6 years, with people in the older than 30 year age group giving higher (lower) ages on average.
We have very strong evidence that adults who are older than 30 years of age gave, on average, a higher age of the person in the image (p = 0), by somewhere between 2 and 9 years higher (with 95% confidence). This result appears to confirm our research question.
ONE SAMPLE T-TEST Q~ On average, are audiobooks on LibriVox longer or shorter than 840 mins?
The p-value is very small (large) and can be reported as p < 0. We have very strong evidence that the mean length for all audiobooks on LibriVox is not 840 minutes. With 95% confidence the mean length for all audiobooks on LibriVox is somewhere between 287 and 510 minutes.
The hypothesised value (840 minutes) is not (is) one of the plausible values in the confidence interval which is consistent with the p-value being less (more) than 5%.
Because this data was a random sample of audiobooks from the LibriVox website we can (cannot) generalise these findings to all audiobooks on LibriVox. We cannot (can) also generalise this to the audiobooks available on the library app.
Chapter 8 (Explanation)
Sample Size a researcher originally had a sample of 400 to construct a confidence interval, but now only has a sample of 100.
The Size of the sample has decreased. This means that the researcher will have less data, meaning the confidence interval will be wider, as it will be harder for them to precisely estimate the parameter of interest.
Two Sample T-Test - Observational Study to see if breastfeeding impacted GCI levels
This type of study is an observational study because whether or not the infant was breastfed was determined by the parents before the study.
The mean GCI for the children who were breastfed as infants is 4 higher than the mean GCI for children who were not breastfed.
Null Hypothesis: The underlying mean GCI scores are the same for both the breastfed and Not Breastfed children. two-sided alternative hypothesis: The underlying mean GCI scores for the breastfed children are not the same for the Not Breastfed children.
The p-value for this test is 0. This tells us we have some evidence that there is a link between breastfeeding in infancy and GCI at age 4 95% confidence, we estimate that the underlying mean GCI score of the breastfed children is somewhere between 0. and 7 higher than that of the not-breastfed babies.
The type of conclusion we could make from this study is a claim about a link between variables (experiment-to-causation inference or sample-to-population inference)
In order to claim that breastfeeding in infancy affects GCI at 4 years of age you would need data from a study where infants were randomly allocated to be either breastfed or not breastfed. This process would help to balance out possible confounding variables, such as parents’ education or income level, across the two groups.
The scope of inference for this study is that we could generalise these findings to all 4 year olds similar to those in the study.
Study Design Causal claim is a claim of cause and effect. See if the experiment was well designed and carried out or if other confounding variables may be at hand.
Chapter 9 (Variance)
Test Statistics vs P Value Q~Do different pre-run cold drinks give different run-to-exhaustion times?
The data above is a rectangular data set because each row is a different participant, and each column is a different variable measure for each participant.
The difference variable contains positive values for all participants, which means that the difference was calculated as ice_slurry - cold_water, as the time taken to run to exhaustion was higher for ice_slurry compared to cold_water for all participants.
The two variables, ice_slurry and cold_water, were made on each participant, so we do not have two independent samples, we have one sample. Data with this design are called paired data when making comparisons.
The p-value for the hypothesis test is 0 and indicates that there is some-to-strong evidence that the respiration (gill) rate changes with the amount of calcium in the water.
It can be claimed that the amount of calcium in each tank caused the changes in respiration rates for these fish, because the study was a well-designed and executed experiment.
Two of the pairs of groups have differences that are significant at the 5% level.
With 95% confidence, the underlying mean gill rate for a low level of calcium is somewhere between 0 and 18 beats per minute higher than the underlying mean gill rate for a medium level of calcium.
With 95% confidence, the underlying mean gill rate for a high level of calcium is somewhere between 9 lower and 8 higher compared to than the underlying mean gill rate for a medium level of calcium.
ANOVA (F-TEST) - Observational Does the time taken to travel the last 50 metres before a set of traffic lights depend on age group?
The Older than 40 age group has the highest mean time to travel the last 50 metres.
The sample mean for the 30 to 40 group is higher than the Less than 30 group. We can not claim that the underlying mean time to stop is higher for the 30 to 40 group as the comparison intervals overlap. There seems to be a bit more variation in the 30 to 40 year old group compared to the other two groups.
The ratio of the largest standard deviation to the smallest standard deviation is 1, which indicates that the groups have similar variation.
The overall risk of an arrest is 48%
Of the incidents involving drugs and/or alcohol, the proportion that resulted in an arrest is 0. This is an estimate of the risk of arrest given that the incident involved drugs and/or alcohol. Of the incidents involving damage to property, the proportion resulting in an arrest is 0. This is an estimate of the risk of arrest given that the incident involved damage to property.
For these data, the proportion of incidents that resulted in an arrest is not the same as the proportion of incidents involving drugs and/or alcohol that resulted in an arrest. This suggests that whether an incident results in an arrest or not is dependent on the type of incident. For an incident that involved drugs and/or alcohol, an estimate of the risk of arrest was 7. times the risk for an incident that involved damage to property. This is an estimate of the relative risk.
Relative risk calculation: Dive the two values together.
Chi-Square Test Q~ Does the injury rate of a trampoline change with different seasons?
Trampoline Injuries Occurred mostly in spring.
While the proportion of trampoline injuries in each season is not identical in our sample data a test for equal proportions was carried out to investigate if this difference in proportions could be explained by sampling variability or if it is an indication of a true difference in proportions.
Under the null, we would expect the proportion of injuries in each season to be 25% (p = 0). Since we have a total of 353 injuries in our data, we have enough data to carry out a chi-square test.
STAT108 - Interactive Exercises Notes (5-11)
Course: Statistics for Commerce (STATS108)
University: University of Auckland
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