Calculating Expected values and Chi Squared Values

Carolee Pederson
23 Apr 201413:00
EducationalLearning
32 Likes 10 Comments

TLDRThis video script offers an educational walkthrough on calculating expected values and the chi-squared statistic in a hypothesis testing scenario. The example involves surveying 60 individuals, divided by age, to determine if those over 21 are more likely to pass their driver's test on the first attempt. The script guides viewers through the process of tallying observed values, calculating expected frequencies using the formula (row total * column total) / total sample size, and then deriving the chi-squared value by comparing observed and expected results. The chi-squared value, obtained through a series of calculations, will be utilized in subsequent videos to assess the significance of the observed age-passing rate relationship.

Takeaways
  • 📚 The video introduces the concept of chi-squared testing and its terminology and formulas.
  • 🔍 The example involves surveying 60 people to determine if age over 21 correlates with passing a driver's test on the first attempt.
  • 👥 The sample consists of 40 people over 21 and 20 people under 21, with results categorized by pass/fail for each age group.
  • 📊 The observed values are recorded, showing 36 people over 21 passed, 12 under 21 passed, 4 over 21 failed, and 8 under 21 failed.
  • 🧮 The expected values are calculated using the formula: (row total * column total) / total number of people, to determine what should happen in a normal distribution.
  • 📝 The expected values calculated are 32 over 21 passing, 16 under 21 passing, 8 over 21 failing, and 4 under 21 failing.
  • 📉 The observed and expected values are compared to identify any significant differences that may suggest a non-random relationship between age and test results.
  • 🔢 The chi-squared value is calculated using the formula: (O - E)^2 / E, where O is the observed value and E is the expected value.
  • 📈 A contingency table is set up to organize the observed and expected values for the chi-squared calculation.
  • 📚 The chi-squared value obtained, 7.5 in this example, will be used in further analysis to test the hypothesis regarding the relationship between age and passing the driver's test.
Q & A

  • What is the purpose of the chi-squared test introduced in the video?

    -The purpose of the chi-squared test is to determine if there is a significant association between two variables, in this case, age and the likelihood of passing a driver's test on the first attempt.

  • What are the steps for calculating expected values in a chi-squared test as described in the video?

    -The steps for calculating expected values involve multiplying the row total by the column total and then dividing by the total number of people in the sample.

  • How many people over 21 were surveyed in the example provided in the video?

    -In the example, 40 people over the age of 21 were surveyed.

  • How many people under 21 were surveyed in the example provided in the video?

    -In the example, 20 people under the age of 21 were surveyed.

  • What is the observed number of people over 21 who passed their driver's test the first time in the example?

    -The observed number of people over 21 who passed their driver's test the first time is 36.

  • What is the observed number of people under 21 who passed their driver's test the first time in the example?

    -The observed number of people under 21 who passed their driver's test the first time is 12.

  • What is the expected number of people over 21 who should have passed their driver's test the first time according to the video?

    -The expected number of people over 21 who should have passed their driver's test the first time is 32.

  • What is the expected number of people under 21 who should have passed their driver's test the first time according to the video?

    -The expected number of people under 21 who should have passed their driver's test the first time is 16.

  • How is the chi-squared value calculated in the video?

    -The chi-squared value is calculated by taking the difference between the observed and expected values, squaring the result, and then dividing by the expected value.

  • What is the calculated chi-squared value in the example provided in the video?

    -The calculated chi-squared value in the example is 7.5.

  • What does the chi-squared value represent in the context of the test?

    -The chi-squared value represents the extent of the difference between the observed data and the expected data under the null hypothesis. A higher chi-squared value suggests a greater difference and potentially a significant association between the variables.

Outlines
00:00
📊 Introduction to Chi-Squared and Expected Values

The video begins with a recap of the introduction to chi-squared tests, terminology, and formulas. The focus then shifts to calculating expected values for a sample and the chi-squared value. An example is presented involving observing 60 people and asking whether they passed their driver's test the first time, categorized by age over or under 21. The observed data is collected and organized into a contingency table. The video emphasizes not to worry about the seven steps of hypothesis testing yet, as they will be covered in later videos. The audience is encouraged to write down the example and use a PDF for reference to save time.

05:05
🧮 Calculating Expected Values and Chi-Squared

This paragraph explains the process of calculating expected values in a normal distribution. The observed totals for passing and failing the driver's test are calculated, and then the expected values are computed using the formula: (row total * column total) / total number of people. The expected values for each age group, both for those who passed and failed, are calculated. The video shows the calculations for each cell in the contingency table, resulting in expected values of 32, 16, 8, and 4 for the respective categories. These expected values are then used to calculate the chi-squared value, which is a key component in hypothesis testing to determine if there is a significant relationship between age and passing the driver's test on the first attempt.

10:06
🔢 Chi-Squared Calculation and Interpretation

The final paragraph details the calculation of the chi-squared value using the observed and expected values from the previous step. The formula for chi-squared is demonstrated, which involves taking the difference between observed and expected values, squaring it, and then dividing by the expected value. Each cell in the contingency table is processed this way, and the results are summed to obtain the chi-squared value. The example provided results in a chi-squared value of 7.5. This value will be used in subsequent videos to perform hypothesis testing and determine the statistical significance of the observed data regarding the relationship between age and passing the driver's test on the first attempt.

Mindmap
Keywords
💡Chi-squared
Chi-squared is a statistical method used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in a study. In the video, chi-squared is introduced as a tool to analyze the relationship between age and the likelihood of passing a driver's test on the first attempt. The script goes through the process of calculating expected values and the chi-squared value to test the hypothesis.
💡Expected Values
Expected values are the frequencies that would be predicted if the null hypothesis were true. They are calculated using the formula: (row total * column total) / total number of observations. In the context of the video, expected values are used to determine what the passing rates for both age groups (over and under 21) should be in a normal distribution, based on the total number of people surveyed.
💡Observed Values
Observed values are the actual frequencies or counts that are collected during the data collection process. In the video, the observed values are the actual numbers of people over and under 21 who passed or failed their driver's test. These values are compared with the expected values to calculate the chi-squared statistic.
💡Hypothesis Testing
Hypothesis testing is a statistical method that is used to make decisions about a population parameter using sample data. The video script mentions that after calculating the expected and observed values, hypothesis testing will be conducted to determine if there is a connection between age and the ability to pass a driver's test on the first attempt.
💡Contingency Table
A contingency table is a type of table in a matrix form that displays the frequency distribution of variables. In the video, a contingency table is used to organize the observed and expected values for people over and under 21 who passed and failed their driver's test. This table is essential for calculating the chi-squared value.
💡Normal Distribution
Normal distribution, also known as Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data points close to the mean are more frequent in occurrence. The video discusses calculating expected values based on the assumption of a normal distribution, which helps in determining what the passing rates 'should' be.
💡Chi-squared Value
The chi-squared value is the result of a statistical test that measures how likely it is that the observed data occurred by chance if the null hypothesis were true. In the script, the chi-squared value is calculated by comparing the observed and expected values, and it will be used in subsequent videos to make inferences about the relationship between age and test performance.
💡Null Hypothesis
The null hypothesis is a statement of no effect or no difference, which is what hypothesis testing aims to reject or fail to reject. In the context of the video, the null hypothesis would be that there is no significant difference in the passing rates of the driver's test between people over and under 21.
💡Sample
A sample is a subset of a population that is used for statistical analysis. In the video, the sample consists of 60 people who were asked about their age and whether they passed or failed their driver's test. This sample is used to gather data and perform the chi-squared test.
💡Statistical Significance
Statistical significance refers to the likelihood that an observed difference or effect is real and not due to chance. The video script discusses using the chi-squared test to determine if the difference in passing rates between the two age groups is statistically significant.
Highlights

Introduction to chi-squared and related terminology and formulas.

Explanation of calculating expected values in a sample for chi-squared analysis.

Presentation of an example involving 60 people and their driver's test outcomes.

Collection of data on pass/fail rates for people over and under 21 years old.

Totals calculation for people who passed and failed the test, regardless of age.

Summation of the number of people surveyed in each age group.

Hypothesis testing setup to determine if age affects passing the driver's test.

Assignment instructions to calculate expected values using a specific formula.

Description of the formula for calculating expected values in a chi-squared test.

Calculation of expected values for each age group and pass/fail category.

Comparison of observed values with expected values for the chi-squared test.

Detailed calculation steps for the chi-squared value using observed and expected data.

Presentation of the contingency table for organizing observed and expected values.

Final calculation of the chi-squared value and its significance in statistical analysis.

Explanation of the chi-squared value's role in hypothesis testing and further analysis.

Instruction to write down the entire slide for reference and clarity.

Transcripts

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Chi-squared TestStatistical AnalysisExpected ValuesHypothesis TestingData InterpretationEducational ContentVideo TutorialSample CalculationStatistical FormulasResearch Methodology