Suppose that each observation in a random sample of 100...

Name: Problem 16, Chapter 14: Statistics Learning From Data
Uploaded: 2023-06-26T18:47:00-08:00
Duration: 5 min 32 s
Channel: Jon Southam
Description: Suppose that each observation in a random sample of 100 fatal bicycle accidents in 2015 was classified according to the day of the week on which the accident occurred. Data consistent with information on the Insurance Institute for Highway Safety website (www.iihs.org/iihs/topics/t/generalstatistics/fatalityfacts/overview-of-fatality-facts) are given in the following table.

Suppose that each observation in a random sample of 100 fatal bicycle accidents in 2015 was classified according to the day of the week on which the accident occurred. Data consistent with information on the Insurance Institute for Highway Safety website (www.iihs.org/iihs/topics/t/generalstatistics/fatalityfacts/overview-of-fatality-facts) are given in the following table.

Key Concepts

Random Sampling

Random sampling is a fundamental concept in statistics where each observation in the population has an equal chance of being selected. This approach helps to ensure that the sample is representative of the broader population, reducing bias and allowing for generalizable conclusions from the analysis.

Categorical Data

Categorical data refers to information that can be sorted into distinct groups or categories. In this context, data is classified by the day of the week, making it a nominal variable which is used to record observations based on attributes without any inherent numerical value.

Frequency Distribution

A frequency distribution is a summary of how often different values or categories occur in a dataset. By tabulating the number of fatal accidents for each day of the week, one creates a frequency distribution that can be used to analyze patterns or test hypotheses about the underlying data.

Goodness-of-Fit Testing

Goodness-of-fit tests, such as the chi-square test, are used to compare the observed frequencies in the categories with expected frequencies under a specific hypothesis. This concept is crucial when assessing whether the distribution of accidents across the days of the week matches a theoretical or uniform distribution.

Transcript

00:01 Here you have some data which are the number, is a random sample of 100 bicycle accidents.

00:07 And this is the number of accidents that occurred on a given day of the week, sunday, monday through saturday.

00:13 So 14 on sunday, 13 on monday, and so on and so forth.

00:16 And there's a total of 100.

00:17 So if we add all those up, we see we get 100.

00:20 And what we want to know, if based on these data, is it reasonable to include that the proportion of accidents is not the same for all days of the week? so they're not equal.

00:31 So we can see there's some difference, right? so tuesday has 12, whereas friday has 17.

00:37 And so they're different, but is it significant enough to say they're not all the difference? so this is going to be a perfect chi -squared test for goodness of fit.

00:47 So chi -squared goodness of fit test.

00:51 And we're going to test this at the alpha of 0 .05 level of significance.

01:03 And what that means is we have to set up our hypotheses, but we're going to reject the null hypothesis if our p -value is less than this alpha.

01:10 So our null hypothesis would be that, essentially, this is going to be uniform distribution, that it's uniform.

01:17 So a uniform would be a good fit.

01:20 So all equally likely, all days equally likely.

01:23 All days distribution is a uniform distribution.

01:38 The alternative, all days are not equally likely, not equally likely.

01:48 In other words, not a good fit.

01:49 That's why it's called a goodness of fit test.

01:52 So we get our chi -squared statistic, which is found by taking the sum of all the observed values minus the expected values squared all over the expected values.

02:05 And we have the observed values here.

02:07 Those are our expected counts.

02:08 Or excuse me, those are observed counts right here.

02:11 Observed.

02:12 Now we need the expected counts.

02:15 And the expected counts will be, all right, if it is evenly distributed, if they are all equally likely, then we'd evenly split the 100 across the seven days.

02:26 So what we do is we take 100 divided by seven and assign that to each day.

02:30 And that is our expected amount.

02:35 So there we go.

02:35 Those are the expected counts.

02:36 They're all the same.

02:37 That would mean uniformly distributed, which is what we want to see.

02:42 Now we get our statistic by taking the observed minus the expected squared divided by the expected value.

02:50 So that's what this column of our spreadsheet is filled up with.

02:54 So just to give you a little idea of how this works...