Question

also suppose that the mean amount of financial aid received by students at the college is $1,500 ( μ-15, 00). What is the population? Whatis the variable? What is the population mean? Question: Howaccurate arerandom samplesatprediction thispopulationproportion of0.60? Consider the following random samples and determine the mean amount of financial aid received by the students Random Sample 1 Random Sample 2 Random Sample 3 Sample Means: Each random sample came from a population for which the mean amount of financial aid received by individual students is $1,500. Do all samples have the same means? Do all samples give good estimates of the population mean? Explain. Aparameterisanumberthatdescribes a population. A statisticisanumberthatwe calculate froma sample Find the parameter and the statistics in part I. Find the parameter and the statistics in part li. Complete the table using the statistics symbols. Population) Parameter (Sampio) Ssiatistic Proportion Mean Standard Deviation

Answer 1

What is population? The 200 students of small college with size N=200 is the population.

What is the variable? The amount of financial aid received by individual student at the college is (quantitative) variable (say X).

What is the population mean? The mean amount of financial aid received of $1500 by the students at the college is the population mean. It is denoted by: $\mu = 1500$

Formulae:

Sample size (n)=total number of observations/values in the sample. Here, n=8.

x_i denote sample observations.

Sample mean= sum of all the values of variable present in the sample divided by the sample size.

Sample standard deviation is the square root of sample variance which is computed by-

$\widehat{\sigma }\, = \sqrt{\frac{\sum (x_{}i-\bar{x})^2}{(n-1)}}$

Student's t-statistic that follows t-distribution with n-1 degrees of freedom is given by-

$t\, = \frac{ (\bar{x}-\mu )}{\frac{\widehat{\sigma }}{\sqrt{(n-1)}}} \, \sim \, t_{n-1}$

Null Hypotheisis

H0: $\mu =1500$

Alternative Hypothesis

H1: $\mu \neq 1500$

TABLE	Random sample 1	Random sample 2	Random sample 3
Sample Means	$\bar{x}=16500/8=2062.5$	$\bar{x}=10600/8=1325$	$\bar{x}=5500/8=687.5$
Sample standard deviation	1370.023	1265.758	984.7951
absolute value of student t-statistic	1.086284	0.36579	2.18286
tabulated value for t at 5% level of significance	2.37	2.37	2.37
Result	Accept H0	Accept H0	Accept H0

No, all three samples do not have the same means.

No, all samples do not give good estimates of population mean if we just observe the sample means in the table above. But by procedure if we carry out t-test for single mean, the result completely alters; we accept H0 in each case which means they give good estimates. This is due to small sample size, 8.

Population mean ($\mu$) is the parameter whereas sample mean ($\bar{x}$) is the statistic. Sample standard deviation is also statistic.

Population mean, population standard deviation, population proportion are all parameters.

Sample mean, sample standard deviation, sample proportion, student's t-statistic (used for testing of hypothesis) are all statistic.

Completed table using the statistics symbols is as follows:

	Population (Parameter)	Sample (Statistic)
Proportion	p	$\widehat{p}$
Mean	$\mu$	$\bar{x}$ or $\widehat{\mu }$
Standard Deviation	$\sigma$	$\widehat{\sigma }$

Note that symbols may (or may not) change according to domain of application of statistics. Sometimes, all capital letters are used for population "parameter" and small letters for sample "statistic".