Sampling distribution of a statistic

In summary, if you integrate between 1 and 3 with the different limits, you get $\frac{4}{3}$, but if the density is 1 between 1 and 2, you get 0.
  • #1
das1
40
0
Looking at another textbook problem, hope someone can let me know if I'm on the right track:

Let $X_1, X_2, ... X_{25}$ be a random sample from some distribution and let $W = T(X_1, X_2, ... X_{25})$ be a statistic. Suppose the sampling distribution of W has a pdf given by $f(x) = \frac{2}{x^2}, 1 < x < 2$ . Find the probability that W < 1.5

So from what I understand, we're sampling 25 numbers from a population and then getting some statistic, like a mean or a median from those 25 numbers. If you sample enough times, you get $f(x) = \frac{2}{x^2}, 1 < x < 2$ as a sampling distribution for that statistic, between 1 and 2. I'm pretty sure you need to take a definite integral here with 1.5 as an upper limit, but what's the lower limit? 1? 0? $-\infty$ ? Also, do you integrate with respect to x or some other variable? Or maybe I'm way off. Hope someone can help, thanks!
 
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  • #2
das said:
Looking at another textbook problem, hope someone can let me know if I'm on the right track:

Let $X_1, X_2, ... X_{25}$ be a random sample from some distribution and let $W = T(X_1, X_2, ... X_{25})$ be a statistic. Suppose the sampling distribution of W has a pdf given by $f(x) = \frac{2}{x^2}, 1 < x < 2$ . Find the probability that W < 1.5

So from what I understand, we're sampling 25 numbers from a population and then getting some statistic, like a mean or a median from those 25 numbers. If you sample enough times, you get $f(x) = \frac{2}{x^2}, 1 < x < 2$ as a sampling distribution for that statistic, between 1 and 2. I'm pretty sure you need to take a definite integral here with 1.5 as an upper limit, but what's the lower limit? 1? 0? $-\infty$ ? Also, do you integrate with respect to x or some other variable? Or maybe I'm way off. Hope someone can help, thanks!

Hi das,

I interpret the specification of $f$ to mean that the density is 0 outside of the given interval.
What do you get if you integrate $f$ between $1$ and $2$?
 
  • #3
Hi! Thank you you've been very helpful.
Integrating between 1 and 2 gets the indefinite integral of $-\frac{2}{x}$. Between 2 and 1 this is (-1) - (-2) = 1. Which makes sense, but how do I take that info and figure out the probability that W < 1.5 ?

Thanks again
 
  • #4
das said:
Hi! Thank you you've been very helpful.
Integrating between 1 and 2 gets the indefinite integral of $-\frac{2}{x}$. Between 2 and 1 this is (-1) - (-2) = 1. Which makes sense, but how do I take that info and figure out the probability that W < 1.5 ?

Thanks again

It confirms that the density is indeed 0 outside the interval.
And it means that:
$$P(W<1.5) \underset{def}{=} \int_{-\infty}^{1.5} f_W(x) \,dx = \int_{1}^{1.5} f(x) \,dx$$
 
  • #5
But if the density is 1 between 1 and 2, doesn't that mean that it should be 0 everywhere else? And that's not true--If I integrate with different limits, say between 1 and 3, I get $\frac{4}{3}$. If the density were 0 outside this interval, doesn't that mean I should still get 1 between 1 and 3?
 
  • #6
das said:
But if the density is 1 between 1 and 2, doesn't that mean that it should be 0 everywhere else? And that's not true--If I integrate with different limits, say between 1 and 3, I get $\frac{4}{3}$. If the density were 0 outside this interval, doesn't that mean I should still get 1 between 1 and 3?

You are correct that the density is 0 elsewhere. So if you did something like: \(\displaystyle \int_{1}^{3}f(x) \,dx\) you would have to split it up into two integrals since $f(x)$ is not the same over that region.

\(\displaystyle \int_{1}^{3}f(x) \, dx = \int_{1}^{2} \frac{2}{x^2} \, dx+\int_{2}^{3}0 \, dx\)
 
  • #7
Ah ok thank you. Guess there's no reason you'd ever need to complicate your life by doing that.
 

Related to Sampling distribution of a statistic

What is a sampling distribution of a statistic?

A sampling distribution of a statistic is a theoretical distribution that shows all possible values of a statistic that could be obtained from repeated random samples of a certain size from a population. It is used to understand the variability of a statistic and make inferences about the population.

Why is a sampling distribution of a statistic important?

A sampling distribution of a statistic is important because it allows us to make inferences about a population based on a sample. It also helps us understand the accuracy and variability of a statistic, which is crucial in hypothesis testing and making decisions based on data.

How is a sampling distribution of a statistic different from a population distribution?

A population distribution shows the distribution of a variable in the entire population, whereas a sampling distribution of a statistic shows the distribution of a statistic (such as mean or proportion) in all possible samples of a certain size from the population. In other words, a population distribution is the true distribution of a variable, while a sampling distribution is an approximation based on samples from the population.

What factors affect the shape of a sampling distribution of a statistic?

The shape of a sampling distribution of a statistic is affected by three main factors: sample size, population distribution, and sample selection. As the sample size increases, the sampling distribution becomes more symmetrical and approaches a normal distribution. The population distribution also plays a role, as a skewed population distribution will result in a skewed sampling distribution. Lastly, the method of sample selection (e.g. random sampling, convenience sampling) can also impact the shape of the sampling distribution.

How is the central limit theorem related to the sampling distribution of a statistic?

The central limit theorem states that the sampling distribution of a statistic (such as the mean) will be approximately normal if the sample size is large enough, regardless of the shape of the population distribution. This is why the normal distribution is often used to approximate sampling distributions in statistical analyses.

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