Light Internet users have session times that are...

Light Internet users have session times that are exponentially distributed with mean 2 hours, and heavy Internet users have session times that are exponentially distributed with mean 4 hours. * (a) Use the Neyman-Pearson method to find a hypothesis test to determine whether a given user is a light user. Design the test for $\alpha=5 \%$. (b) What is the probability of detecting heavy users?

Light Internet users have session times that are exponentially distributed with mean 2 hours, and heavy Internet users have session times that are exponentially distributed with mean 4 hours.
* (a) Use the Neyman-Pearson method to find a hypothesis test to determine whether a given user is a light user. Design the test for $\alpha=5 \%$.
(b) What is the probability of detecting heavy users?

Key Concepts

Neyman-Pearson Lemma

The Neyman-Pearson lemma provides the foundation for finding the most powerful test for a simple hypothesis against a simple alternative. In statistical hypothesis testing, it involves comparing the likelihood ratio of two probability density or mass functions and setting a threshold such that the probability of a Type I error (rejecting a true null hypothesis) does not exceed the chosen significance level.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. It is characterized by its rate parameter, which is the reciprocal of the mean, and has the memoryless property. This distribution is commonly used to model lifetimes or waiting times, among other phenomena.

Likelihood Ratio Test

The likelihood ratio test is a method based on comparing the ratio of the likelihoods under two competing hypotheses. By evaluating this ratio against a critical value determined by the desired significance level, the test decides which hypothesis is more strongly supported by the data. It is especially useful in scenarios where the data likelihoods can be explicitly computed.

Significance Level (Type I Error)

The significance level, often denoted by alpha, is the probability threshold for committing a Type I error, which occurs when the null hypothesis is rejected even though it is true. Setting this level controls the risk of making such an error and dictates the rejection region of the hypothesis test.

Power of a Test

The power of a test is the probability that it correctly rejects the null hypothesis when the alternative hypothesis is true. It reflects the test's ability to detect an effect, with higher power indicating a greater chance of identifying a true effect or difference. This concept is essential in evaluating the effectiveness of a statistical test.

Hypothesis Testing

Hypothesis testing is a structured methodology for making decisions about population parameters based on sample data. It involves formulating a null hypothesis against an alternative hypothesis, selecting an appropriate test statistic and decision rule, and making inferences while controlling for error rates. This process underpins many statistical analyses in various fields.

Transcript

00:01 Hi, so for the given question, the question is given as, that's my personal distribution is skewed to the right.

00:12 You know, take 100 simple random samples, similar 100 simple random sample size n equal of 10.

00:16 Population mean and 4x7 division is 5 minutes each.

00:20 Everybody data here in c3, so in c3 i've taken the data, which is skewed towards the right, right.

00:28 100 random samples here we have which are skewed towards the right here we have that is for c3 and c4 i have standardized it as i have sanitized it so i again get the histogram for a hundred number simple random samples that is skewed towards the right you can see the skewed towards the right population mean standardized that's going out to be zero we have and it is skewed to the right the sample uh data now after after that, it's a two tail.

01:03 So if you look at the a part, simulate obtaining a simple random sample that we have done.

01:10 So that is simulate obtaining a simple random sample of 10 individuals waiting in line, where the wait time is expected to be five minutes.

01:19 So we have just simulate that.

01:22 Now, we have just test the nullipo.

01:24 This is where we have mu equal to 5 and mu not equal to 5.

01:32 So for that, mu equal to 5, we equal to 5, not equal to 5, we have to go for two tail p value.

01:40 Next we have done.

01:44 Two tailed p value is given here.

01:48 In 3 7, that's a total p value for each under random samples.

01:53 In 3 7, we have two tail p values, right? so now, let's move to the next part with the question...

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