Who tweets? Do younger people use Twitter more often than...

Who tweets? Do younger people use Twitter more often than older people? In a random sample of 316 adult Internet users aged 18 to $29,26 \%$ used Twitter. In a separate random sample of 532 adult Internet users aged 30 to $49,14 \%$ used Twitter. ${ }$ (a) Calculate the standard error of the sampling distribution of the difference in the sample proportions (younger adults - older adults). What information does this value provide? (b) Construct and interpret a $90 \%$ confidence interval for the difference between the true proportions of adult Internet users in these age groups who use Twitter.

Who tweets? Do younger people use Twitter more often than older people? In a random sample of 316 adult Internet users aged 18 to $29,26 \%$ used Twitter. In a separate random sample of 532 adult Internet users aged 30 to $49,14 \%$ used Twitter. ${ }$
(a) Calculate the standard error of the sampling distribution of the difference in the sample proportions (younger adults - older adults). What information does this value provide?
(b) Construct and interpret a $90 \%$ confidence interval for the difference between the true proportions of adult Internet users in these age groups who use Twitter.

Key Concepts

Critical Value and Confidence Level

The critical value is a multiplier that corresponds to the desired level of confidence (such as 90%), typically taken from a standard normal or other relevant distribution. It is used in constructing the confidence interval by accounting for the extent of potential variation in the estimate, thus ensuring that the interval captures the true parameter with the specified probability.

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is used to estimate an unknown population parameter with a certain level of confidence. It reflects the uncertainty associated with the sample estimate and provides a probabilistic assessment of where the true parameter value is likely to lie.

Sampling Distribution of the Difference in Proportions

This concept refers to the probability distribution of the difference between two sample proportions, obtained when random samples are taken from two populations. It provides the framework for understanding how the difference between sample estimates might vary from one sample to another and is crucial for making inferences about the difference between the true population proportions.

Standard Error

The standard error is a measure of the variability or dispersion of a sampling statistic, such as the difference between two sample proportions. It quantifies the expected deviation of an estimate from the true population parameter due to sampling variability, thereby serving as a key ingredient in constructing confidence intervals and hypothesis tests.

Transcript

00:01 In this video, we are going to focus on the information given about younger people using tudor more often than the older people.

00:08 So based on the given information, let us calculate the standard error first.

00:12 So here the sample size is equal to 316 and the sample size of the second group is 532 and the proportion for first group is 26 percentage which can be written as 0 .26 and then for the second group p2, p2 value is 14 percentage which can be written as 0 .14.

00:35 And now here it can be said that standard error of the sampling distribution of the difference in sample proportion is the standard deviation of the sampling distribution to of the difference in sample proportion.

00:52 And here we can mention one thing, then we will write the formula for standard deviation.

00:57 And it is given that we have a sample of 3 .1.

00:59 Adult internet users, right? so it can be written over here as such.

01:06 And this means that these internet users are aged between 18 to 29.

01:14 And here we can say that this is less than 10 % of all the 18, we can say 10 % of all the adult users.

01:24 So it will be here that it is less than 10 % percent.

01:29 Of all adult internet users of all adult users.

01:39 And it can be written like this.

01:41 Now here we can mention that the sample of 75 adult internet users aged between, basically they have the age of 32, it will be 49, and this is also less than 10 % of all internet users aged between, what the age? so, it is basically less than 10 % of all internet users.

02:08 Let us write internet adult users basically.

02:13 So it should be here adult.

02:16 And then here it is going to be users aged aged 30 to 49 years.

02:28 So, hence, from this very information, we can write the value of standard deviation as p1 times 1 minus p1 divided by n1 and then it will be here plus p2 times 1 minus p2 divided by n2 and from this very information we can say that sigma value is here going to be equals to so if we plug in the values over here it will be 0 .26 and then it will be here times 1 minus 0 .26 and then it is here plus 0 .14 and then it will be here 1 minus 0 .14 and in the denominator we are going to have 532 and then it will be here we can say 5 not 5 it will be 316 and hence from this very information sigma value will be here equals to let us write the values it is basically like this and here it will be 0 .26 times 0 .74 and in the denominator it will be 316 plus it is here going to be 0 .14 times 0 .86 and in the denominator it will be 532 and it is going to give us the value which is approximately equals to let us write so this is going to be the standard deviation value right so it is 0 .0289 and thus this standard error describes.

04:06 This is the standard error and this is the basically standard deviation also because we have already said that standard error of the sampling distribution of the difference in sample proportion is the standard deviation of the sampling distribution of the difference in sample proportion.

04:21 So we can say that this standard error describes that we expect the difference in sample proportion to vary about, let us write vary by about to basically it is 0 .02 89 from the mean difference.

04:43 So it can be written over here like this and this will be our answer for the first part of this very problem.

04:53 So here the answer will include this very conclusion as well as this very numerical value.

04:59 So we can put this thing inside a box because this is our answer.

05:03 Now let us try to find the answer for the second part of this very problem.

05:08 So in the second part, we can say that we have all these values but we have one more value which is the confidence interval.

05:15 So confidence level basically.

05:17 So confidence level here is 90 percentage.

05:20 That means it will be 0 .0 .90.

05:25 Very sorry for saying it wrong.

05:26 It is 0 .90...

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