H0: The random variable X is binomial with n=4, p=0.8 H1: The random variable X is not binomial

H0: The random variable X is binomial with n=4, p=0.8 H1:...

Key Concepts

Observed and Expected Frequencies

Observed frequencies are the actual counts collected from data, while expected frequencies are the counts predicted under a specified model or hypothesis. Comparing these frequencies allows statisticians to evaluate the accuracy of the model. A large discrepancy between observed and expected frequencies may indicate that the model or hypothesis does not fit the data well, often leading to further investigation or a change in the model assumptions.

Goodness-of-Fit Test

A goodness-of-fit test is a statistical test used to determine how well observed sample data match the expected distribution under a specific hypothesis. One common example is the chi-square goodness-of-fit test, which compares the observed frequencies in various categories with the frequencies expected under a given theoretical distribution. The resulting test statistic helps in assessing whether there are significant deviations that suggest inadequate model fit.

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. It is parameterized by n, the number of trials, and p, the probability of success in each trial. This distribution is used to model scenarios where outcomes can be classified into two distinct categories, such as success or failure.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about population parameters based on sample data. It involves formulating a null hypothesis that represents a status quo or a specific claim, and an alternative hypothesis that suggests a different scenario. Decisions are made by comparing the observed data against what is expected under the null hypothesis, often using a test statistic and a corresponding p-value to determine if the null hypothesis should be rejected.

Transcript

00:01 Okay, what is a null hypothesis that we have? the random variable x is binomial.

00:07 Okay, where n is equal to 4 and p is equal to 0 .8.

00:16 And the alternative hypothesis that we have is that the random variable x is not binomial with n is equal to 4 and p is equal to 0 .8.

00:29 So, null on the alternative hypothesis is already given to us.

00:32 Now let us just look at the table that we have, right? so what is x we have from 0 to 4 0 1 2 3 4 okay what are the observed frequencies the observed values so this is the observed column 138 132 so this is 1 this is 38 this is 138 this is 132 this is 1 32 this is 40 and 380 and 440 and 389 440 and 389 now this one so x is equal to zero looks all of an outlier the expected values are what are the expected values this is 1 .6 25 .6 1 .6 25 .6 153 .6 153 .6 for both of them in the 4 .09 .6 .409 .6.

02:00 All right.

02:02 We already have the null and the alternative hypotheses with us.

02:07 Now, what we have to do is find the kai square statistic.

02:13 How do you calculate the kai square statistic? in this case, we already have the expected value with us.

02:20 So, this makes our calculation a little shorter, not easier, but a little shorter.

02:24 Right otherwise in order to calculate the expected value the formula that is used is the sample size the sample size which happens to be n how do you get n you sum all of these up some all the expected values up and you will get the value of n so if you do this this is 1 plus 389 which is 390 plus 440 is 8 30 plus 38 is 860 plus 38 is 860 66 68 plus 132 is 1000 so this overall is 1000 so this is the sample size multiplied by the probability for probability for each category or i can say for category i in this case because we are calculating ei okay but in this question they have not given us the probability they have directly given us the expected value okay the next step is after getting the expected value, we need to calculate the kai squared statistic.

03:38 How do you calculate this? for every category that you have, you find the difference of observed and expected value, you square it, you divide that value by the expected value, and in the end you sum all of these values up.

03:50 So let us look at this formula over here in action.

03:54 So what is the difference in this case? the difference over here happens to be 0 .6.

04:06 So 0 .6, we square it and divide this by the expected value, which is 1 .6.

04:12 So my answer is 0 .225 in this case.

04:16 0 .225.

04:18 Similarly, the difference over here is 38 minus 25 .6.

04:24 I square this.

04:26 12 .4 square is 153 .76 .76 and i divide this by 25 .6, which is 6 .006.

04:37 This is 6 .006.

04:42 Okay, then the difference is 153 .6 minus 132.

04:51 21 .6, we square this.

04:57 466 .56 and divide this by 153 .6, which is the expected value.