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Inferential Statistics

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Chapter 9 Introduction to the t-statistic

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Episode 12: Inferential Statistics and Hypothesis Testing Playbook

Using a case study on comparing salaries of police and social services department in new york for 2020 and 2021..

This post acts as a handy cheat sheet on how to draw important inferences from a given dataset using statistics and hypothesis testing.

Citywide Payroll Data (Fiscal Year) was taken from “Data.gov” . Each record represents the following statistics for every city employee:

Agency, Last Name, First Name, Middle Initial, Agency Start Date, Work Location Borough, Job Title Description, Leave Status as of the close of the FY (June 30th), Base Salary, Pay Basis, Regular Hours Paid, Regular Gross Paid, Overtime Hours worked, Total Overtime Paid, and Total Other Compensation.

For the purpose of simplicity, I have extracted the base salaries of only two departments i.e. Police department and Social Services for fiscal years 2020 and 2021 and have taken a subset of relevant columns.

• Step 0: Descriptive statistics and their interpretation.

We use the Data Analysis toolpack in Excel to quickly get descriptive stats for the two departments. Refer to sheet “descriptive_stats” in inferential_stats_case_study.xlsx [1].

We make the following observations and come up with relevant questions related to these observations which we will later answer through various statistical tests:

• Is this difference statistically significant?

The police department’s salaries vary a bit more than the social services department as shown by the standard deviation.

• This implies a right/positive skew.
• But the small value for “Skew” for police shows that the tails don’t extend too far to the right.
• Kurtosis less than 3 also shows that the distribution for police’s salaries is thin-tailed and outlier frequency is low.
• Kurtosis is > 3 for Social Services which implies a higher outlier frequency and fat-tailed distribution.

Inferential Statistics:

There are 3 main themes in inferential statistics and 2 subsequent hypothesis tests:

• Population Variance Known (z-test).
• Population Variance Unknown (t-test).
• Confidence Interval for Mean Difference, Dependent Samples: Example: Calculate the change in mean salary of NYPD from one year to the next.
• t-Test: Paired Two Sample for Means: Example: Testing whether change in police salary from one year to the next was statistically significant.
• t-Test: Two-Sample Assuming Equal Variances: Example: Testing whether the difference in police salary and social services salary is statistically significant.

Confidence Interval for Mean, Single Population:

• Step 1(a): If we know the population variance for the salary of each department, calculate the confidence interval for the mean salary of a police officer and a social worker in New York for the year 2020. Assume that the police department’s salary has a population standard deviation of $80k and social services has a population standard deviation of $60k.

Refer to sheet “1. Known variance, CI, z-table” in inferential_stats_case_study.xlsx [1].

Interpretation: We are 95% confident that a police officer's salary was from $67,794 to $69,256 and a social worker's salary was from $57,861 to $60,050 in NYC in 2020.

• Step 1(b): If we do not know the population variance for the salary of each department, calculate the confidence interval for the mean salary of a police officer and a social worker in New York for the year 2020.

Refer to sheet “2. Unkown var, CI, t-table” in inferential_stats_case_study.xlsx [1].

Interpretation: We are 95% confident that a police officer's salary was from $68,219 to $68,830 and a social worker's salary was from $58,511 to $59,400 in NYC in 2020.

Confidence Interval for Mean Difference, Dependent Samples:

• Step 2: We are interested in knowing the salary increase for each department from 2020 to 2021. Calculate the confidence interval for that increase with 95% confidence.

Refer to sheet “3. Two Means Dependent Samples” in inferential_stats_case_study.xlsx [1].

Interpretation: We are 95% confident that there was a salary increase of $2,113 to $2,237 for the police department. The whole interval is positive so we can be sure that the salary has increased. We are 95% confident that there was a salary increase of $439 to $513 for the social services department. The whole interval is positive so we can be sure that the salary has increased.

Confidence Interval for Mean Difference, Independent Samples

• Step 3(a): We are interested in knowing the salary difference between police and social services. Calculate the confidence interval for that difference with 95% confidence. Assume that the police department’s salary has a population standard deviation of $80k and social services has a population standard deviation of $60k.

Refer to sheet “4. Two Mean Ind Samp, known var” in inferential_stats_case_study.xlsx [1].

Interpretation: We are 95% confident that the true mean difference between police department and Social Services department falls in the interval [$8,253 , $10,885] in 2020.

• Step 3(b): We are interested in knowing the salary difference between police and social services. Calculate the confidence interval for that difference with 95% confidence. We do not known the population variance but we know it’s almost equal for both departments.

Refer to sheet “5. Two Mean Ind Samp, unkwn var” in inferential_stats_case_study.xlsx [1].

Interpretation: We are 95% confident that police department has $8-10k higher salaries than social services in 2020.

Hypothesis Testing

T-test: paired two sample for means:.

• Hari Singh is a pessimistic police officer and says there was no significant increase in NYPD salaries from 2020 to 2021. Run a hypothesis test to prove him right or wrong. Run a similar test for Social Services department.

We use t-Test: Paired Two Sample for Means from Data Anlysis Toolpack in Excel.

Refer to sheet “8. Test for mean. Dep samples” in inferential_stats_case_study.xlsx [1].

• We say that the null hypothesis is that the difference in salaries in 2020 and 2021 is less than or equal to $1000.
• A p-value is a probability that the results from the sample dataset are occurred by chance. Low p-values are considered good. Small p-value shows that we can reject the null hypothesis.
• We reject the null hypothesis because t Stat 37 > t critical one-tail of 1.64.
• Hari Singh is wrong. NYPD's salaries have increased from 2020 to 2021.

t-Test: Two-Sample Assuming Equal Variances:

• Christina works in NYC’s social services department and thinks that NYPD is being paid $10k or more on average than SS department. Run a hypothesis test to prove her right or wrong.

We use t-Test: Two-Sample Assuming Equal Variances from Data Anlysis Toolpack in Excel.

Refer to sheet “10. Test for mean, Ind samples” in inferential_stats_case_study.xlsx [1].

Christina is wrong for 2020 but spot for 2021! Social services was paid less than NYPD in 2021 by more than $10k.

Thought of the Week:

During my work as a data analyst, I generally enjoy presenting cool, new data visualizations to stakeholders across diverse departments and love walking them through complicated dashboards and wacthing them have their Aha! moment when they derive useful insights. However, sometimes I have struggled to push back on certain KPI requests that I find rather redundant and have a hard time explaining to executives that adding more metrics to dashboards won’t necessarily be useful.

A few days back, I came across an interesting article titled “Metrics-Focused Data Strategy with Model-First Data Products - Issue #48” by a group of analysts who write for Modern Data 101. They touched upon the concept of “The Metric Dependency Trees”. In their own words, “The objective of the metrics dependency tree is to understand what positively or negatively triggers the targeted metrics and then aid informed actions to pump the metrics as required. In other words, the metrics dependency tree is a brilliant way to instantly find the root causes (RCA) behind business fluctuations and solve them just as quickly. The metric tree also sheds light on the potential of new or enhanced metrics (metric evolution).”

This is something that a lot of us data analysts can keep in mind while engaging with stakeholders! Check out the complete article here [2].

Until next time!

References:

[1] Excel Workbook [2] Metrics-Focused Data Strategy with Model-First Data Products

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Inferential Statistics: Hypothesis Testing

Mar 30, 2019

860 likes | 1.08k Views

Inferential Statistics: Hypothesis Testing. Hypothesis Testing Testing Population Means. Inferential Statistics. Estimation Estimate population means Estimate population proportion Estimate population variance Hypothesis testing Testing population means

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• independent populations
• university reports
• data collection
• rational conclusion
• hypothesis testing process

Presentation Transcript

Inferential Statistics: Hypothesis Testing Hypothesis Testing Testing Population Means

Inferential Statistics • Estimation • Estimate population means • Estimate population proportion • Estimate population variance • Hypothesis testing • Testing population means • Testing categorical data / proportion • Testing population variances • Hypothesis about many population means • One-way ANOVA • Two-way ANOVA

Hypothesis Testing • Hypothesis testing is a scientific method used for making decision, conclusion or prove of the finding of research • Perform on data collected from sample based on stated hypotheses • Hypothesis testing process ensure the reproducibility of result conclusion and the ability to infer the hypotheses on the population with certain confidence level

Hypothesis Testing Process • Forming statistical hypothesis / research hypothesis • Define statistical significant level (α) • Usually 0.05 (5%), or 0.01 for research needing higher accuracy • Select and calculate the appropriate statistic from data collected from sample • Accept/reject hypothesis based on corresponding score tables • Make conclusion / decision

Hypothesis • The expected rational conclusion of statistical analysis • Research Hypothesis • Written in text • Statistical Hypothesis • Written in mathematical form using parameter

Research Hypothesis • Text identifying the expected relation from the research • Based on the variable test • Relational hypothesis • Comparative hypothesis • Based on direction of relationship of variables • Directional hypothesis • Non-directional hypothesis

Relational and Comparative • Relational hypothesis focuses on the relationship between 2 or more variables • E.g. Hours of study is related to examination scores • Comparative hypothesis focuses on comparing the difference between 2 or more variables • E.g. Male students and female students have different exam scores

Directional and Non-directional Directional – able to identify the direction of the relationship between variables Non-directional – unable to clearly identify the direction of the relationship between variables

Research Hypothesis Examples • Handwriting and examination score are related • Relational, non-directional • Handwriting and examination score are positivelyrelated • Relational, directional • Female students get higher final exam score than male student • Comparative, directional • The scores of female and male students are different • Comparative, non-directional

Statistical Hypothesis Written in mathematical form Always use population parameter μ(Mu) = population mean σ (sigma) = Standard Deviation σ2 = variance p = population proportion

Statistical Hypothesis • H0: Null hypothesis • Always non-directional • Must have “=” (also “>=” and “<=”) • H1: Alternative hypothesis • Always directional, exclusively opposite to null hypothesis • Must not have “=” (only “>”, “<”, “!=”) • Be very careful when writing statistical hypothesis, some can be tricky

Statistical Hypothesis Examples Female students get higher final exam score than male student H0 : μf <= μm H1 : μf > μm Female students get final exam score higher than or equal to (no less than) male student H0 : ? H1 : ? The scores of female and male students are different H0 : μf = μm H1 : μf != μm

Error in Hypothesis Testing • Type I Error • Error caused by rejecting H0 when H0 is true • Probability of type I error is equal to α which is statistical significant level defined in the analysis • Type II Error • Error caused by accepting H0 when H0 is false • Probability of type I error is equal to β

Error in Hypothesis Testing

Power of Test • The probability to reject null hypothesis (H0) when it is false. • In other words, the sensitivity to accept alternate hypothesis (H1) when it is true • Power of Test depends on • Sufficient size of sample • The process of data collection • Appropriate selection of statistic according to population distribution and assumption of each statistic

Degree of Freedom • The value that indicates the degree of variability under certain criteria • Degree of Freedom (df) usually bind to the number of sample or groups within sample • E.g. in the estimation of mean, 1 of the sample loses its freedom. Therefore, the df is n-1 • If the mean of 5 data item is 10, and first four data item are 7, 9, 11, 13, the last item must be 10 • Thus, under the calculation of mean, the value of the last item cannot vary, losing its freedom

Hypothesis Testing • Directional Test / One-Tailed Test • Right-Tailed H0 : θ <= θk H1 : θ > θk • Left-Tailed H0 : θ >= θk H1 : θ < θk • Non-directional Test / Two-Tailed Test H0 : θ = θk H1 : θ != θk • θ: A population parameter

Testing Hypothesis of Population Mean

Testing Population Mean Single population Two independent populations Two dependent populations (paired samples)

Steps in Testing Mean • Forming statistical hypothesis from research hypothesis • Left-tailed, Right-tailed, Two-tailed • Define statistical significant level (α) • Usually 0.05 (5%), or 0.01 for research needing higher accuracy • Calculated test statistic • Compare the calculated value to critical value from table to determine if it falls into critical region Right-tailed Left-tailed Two-tailed • Finally, make decision/conclusion

Single Population Mean Known population variance σ2 Unknown population variance σ2, small sample (n < 30) Unknown population variance σ2, large sample (n ≥ 30)

Known Population Variance • Test mean of one sample against a test value • E.g. Test if average total score is more than 55 • Assumptions • Variable of interval or ratio scale • Sampling using probabilistic sampling • Each data item is independent of each other • Known population variance

Known Population Variance Hypotheses Critical Region H0 : μ < μ0 H1 : μ > μ0 σ2 known H0 : μ > μ0 H1: μ < μ0 σ2 known

Example 1 A sawmill cuts wood log into 30-inch lumbers. It is known that the variance of the lumber length is 0.5 inch2. To test the machine, 16 lumbers are sampled yielding the average of 30.25 inch. Assuming normal distribution of lumber population, test if this machine is accurate at significant level 0.05 Hypotheses

Example 1 Calculate test statistic Z-score from table The calculated z-score is -1.96 < 1.41 < 1.96, and does not fall in critical region. Thus cannot reject null hypothesis H0 Accept H0 and reject H1 The machine is accurate at significant level 0.05

Example 2 A research of student proficiency in a university reports that the average score is 100 with SD of 16. A researcher then use a proficiency test on 64 sample students from this university. The result is a mean score of 121. Test if the student proficiency increases at significant level 0.01 Hypotheses α = 0.01

Example 2 Calculate test statistic z-score from table The calculated z-score is 10.50 > 2.326, falling in critical region. Reject H0 and accept H1 The student proficiency increases at significant level 0.01

Unknown Population Variance • Assumptions • Variable of interval or ratio scale • Sampling using probabilistic sampling • Each data item is independent of each other • Unknown population variance

Unknown Population Variance, n<30 Hypotheses Critical Region df H0 : μ < μ0 H1 : μ > μ0 σ2 unknown H0 : μ > μ0 H1 : μ < μ0 σ2 unknown

Example 1 A company believe that its employees will work for the company for no less than 8 year. To test this, 16 sample of employee records are study and are found that they work for the company for 5, 9, 7, 11, 11, 8, 3, 7, 6, 8, 6, 4, 8, 2, 5, 6 years. Is the company’s belief true at significant level 0.05 assuming normal distribution of population. Hypothesis H0: ? H1: ? α = 0.05

Example 1 Calculate test statistic

Example 1 t-score from table The calculated t-score is -2.15 < -1.753, falling in left-tailed critical region. Reject H0 and accept H1 Conclusion?

Example 2 H0 : μ < 17 H1 : μ > 17 A study of complementary class with a sample of 25 students. After the class, the average exam score of the sample is 22 with SD of 5.6. Test if this complementary class can increase the score above the criterion of 17 at significant level 0.05. Hypothesis α = 0.05

Example 2 Calculate test statistic t-score from table: t0.05(24)= 1.711 The calculated t-score is 2.50 > 1.711, falling in right-tailed critical region. Reject H0 and accept H1 The complementary class can increase the exam score above the criterion at significant level 0.05

Unknown Population Variance, n ≥ 30 Hypotheses Critical Region H0 : μ < μ0 H1 : μ > μ0 σ2 unknown H0 : μ > μ0 H1 : μ < μ0 σ2 unknown

Example 1 By sampling 100 female deceased in the previous year, it is found that the average lifespan is 71.8 year with SD of 8.9 year. Does this data support the assumption that the average lifespan of current female citizen is more than 70 years at significant level 0.05? Hypothesis α = 0.05

Example 1 Calculate test statistic z-score from table: The calculated z-score is 2.02 > 1.645, falling in right-tailed critical region. Reject H0 and accept H1 The average lifespan of current female citizen is more than 70 years at significant level 0.05

Example 2 According to a company, the food expense of its employees is 550 THB. In order to adjust salaries, the company want to know if this expense increases. From a sample of 30 employee, the average food expense is 590 THB with SD of 90. (Significant level 0.01) Hypothesis α = 0.01

Example 2 Calculate test statistic z-score from table: Z0.01= 2.326 The calculated z-score is 2.434 > 2.326, falling in right-tailed critical region. Reject H0 and accept H1 The food expense of employees increases at significant level 0.01

Two Independent Populations • Known variances of the two populations • Known • Unknown population variances, small sample (n < 30) • Unknown • Unknown population variances, large sample (n ≥ 30) • Unknown • Unknown population variances but known to be equal

Known Variances of Two Populations • Test mean of one sample against another • Assumptions • Variables (samples) are independent of each other • Variables of interval or ratio scale • Sampling using probabilistic sampling from population of normal distribution • Each data item is independent of each other • Known population variances

Known Variances of Two Populations Hypotheses Critical Region H0 : μ1 – μ2< d H1 : μ1 – μ2 > d σ12 ,σ22 known H0 : μ1 – μ2> d H1 : μ1 – μ2 < d σ12 ,σ22 known

Example H0 : μ1 = μ2OR μ1 -μ2 = 0 H1 : μ1 ≠μ2OR μ1 -μ2 ≠ 0 The quality comparison between golf balls of type A and B is based on drive distance. From the sample of 25 golf balls from each group, the average drive distances of type A and B is 275 yards and 290 yards respectively. If both types have the same population variance of 225, assuming normal distribution of drive distance, test if both types are of the same quality. Hypothesis α= 0.05

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