(a) identify the expected distribution and state H0 and Ha,...

Key Concepts

Interpretation in Context

Interpreting the results involves placing the statistical decision within the framework of the original research question. It explains whether there is sufficient evidence to support the researcher’s claim about the distribution of the data or if the assumption under the null hypothesis remains plausible.

Test Statistic Calculation and Decision Making

Calculating the chi-square test statistic involves summing the squared differences between the observed and expected frequencies divided by the expected frequencies for each category. The resulting statistic is compared to the critical value (or used to compute a p-value) to decide whether to reject the null hypothesis.

Degrees of Freedom

Degrees of freedom in the context of a chi-square goodness-of-fit test are generally calculated as the number of categories minus one. This parameter affects the chi-square distribution's shape and is crucial for determining the appropriate critical value for the test.

Critical Value and Rejection Region

The critical value is the threshold derived from the chi-square distribution corresponding to the chosen significance level. Observing a test statistic that falls in the rejection region (beyond the critical value) leads to the rejection of the null hypothesis, indicating that the observed data significantly deviate from the expected distribution.

Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test evaluates whether the observed frequencies in categorical data match the frequencies expected under a specific distribution. It measures the discrepancies between observed and expected counts and uses a chi-square distributed test statistic to determine if these differences are due to random chance or indicate a poor fit.

Hypothesis Testing

Hypothesis testing is a fundamental procedure in statistics used to assess whether sample data provide enough evidence to reject a default assumption about a population parameter. It involves setting up both a null hypothesis (representing a baseline or no effect) and an alternative hypothesis (representing the claim or effect to be tested), and then using a test statistic and significance level to make an inference about these hypotheses.

Expected Frequencies

Expected frequencies are the counts that one would anticipate in each category based on the assumed distribution under the null hypothesis. They are calculated using the total sample size and the proportions specified by the hypothesis, serving as a benchmark in the chi-square test for comparing against observed frequencies.

Transcript

00:01 All right, so in this question now, a researcher claims that the number of cups of coffee u .s.

00:05 Adults drink per day are distributed according to a certain type of distribution.

00:10 Now, we have been given a figure in which the distribution is given to us.

00:15 Now, what we do is, in order to verify this, we randomly select 1 ,600 us adults and ask them how many cup of coffee they drink.

00:26 So let us just look at the table.

00:28 What are the observations that we have? the first one, the first column, just a moment, yeah, the first one is response.

00:44 So this is 0, 1, 2, 3, 4 or more cups.

00:47 So this is going to be 0 cups, 1 cup, 2 cups, 3 cups, and this is 4 or more cups.

01:07 All right.

01:09 Now what is our observation? the observed values, the observed values are 570, 432, 282, 182 152, 182 152, then we have 164, 164.

01:36 All right.

01:38 Now, what is the distribution that is given to us? now, this is the column for probabilities, the column for probabilities.

01:49 What are the probabilities that are given to us? the distribution is for 0 .0 cups, it's 36%.

01:54 Let me just write this is 0 .36.

01:57 For 1 cup, it is 26 % or 0 .26.

02:01 For 2 cups, it is 19%.

02:02 0 .19.

02:04 For 3 cups, it is 9 % or 0 .9.

02:09 For 4 or more cups, it is 10 % or 0 .1.

02:13 All right.

02:16 What are the null and the alternative hypothesis? the null hypothesis for a goodness of fit test is always that the observed distribution follows the expected distribution so what we can say what is if we look at the wording the number of cups you as coffee adults drink per day are distributed okay the observed distribution the observed distribution of coffee drinkers coffee drinkers and the given distribution and the given distribution are similar.

03:11 What is going to be the alternative for this one? it will be that the observed distribution of coffee drinkers and the given distribution and the given distribution are different.

03:49 Are different.

03:51 All right, now we are going to use the kai square statistic.

03:56 What is the first step in a kai square analysis? the first step is to find the expected values, that is e.

04:06 This is found for all the categories.

04:08 So, e is given by the sample size.

04:12 E is given by the sample size, multiplied by the probability of the category multiplied by the probability of the category of the category all right so what is the sample size the sample size is 1600 this is overall 1600 if i'm not wrong let us just look at this yeah this is 1600 so how do you find the expected values the expected values will be given by probability multiplied by the sample size.

04:59 So this is 1600 multiplied by 0 .36 which is 576 then it is 600 multiplied by 0 .26 which is 416 then it is 1600 multiplied by 0 .19 which is 304 then it is sixteen hundred multiplied by 0 .19 which is 304 then it is 1 ,600 multiplied by 0 .09 which is 14 .34 then it is 1600 144 and then it is 1600 multiplied by 0 .1 which is 160.

05:39 Okay, now that we have the expected values, we see that the sample was random and also that expected values are greater than or equal to 5.

05:47 The next step is to calculate the kaifahua statistic.

05:50 How do we do that? for all the categories, we calculate the difference between observed and the expected values.

05:56 We square them, we divide those by their respective expected values, and in the end we add, all of these up this is going to give me the kai square statistic for my entire problem let us look at this formula in action this column is going to be for the kai square values okay so what did the formula say the difference between the observed and the expected which is 570 minus 576 or yeah so this is going to be 6 we squared it which becomes 36 and divide this by the expected value divided by 576 this is 0 .0625 0 .0 .0 .0 now i do this for all of them so over here it is going to be 16 square which is 256 and divide this by 416...

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