Question

Describe a confidence interval for the difference in means between two population by stating 1. a pair of populations composed of the same type of individuals and a quantitative variable on those populations, 2. sizes and degrees of freedom of samples from those populations, 3. the means of those samples, and 4. the standard deviations of those samples. Then state 5. a confidence level and find 6. find the interval. Finally, perform a test of significance concerning the difference in the means of the populations by stating 7. both a null and an alternative hypothesis and 8. an α-level, then finding 9. the two-sample t-statistic and either 10. rejecting or failing to reject the null hypothesis. Remember that you do not need to list the values of the variable for individuals in either the sample or the population, and that the values for 2, 3, 4, 5, and 8 do not need to be calculated, only stated.

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Answer 1

1)Let \mu _{1} be the population mean of students scores who dont go for mathematics tutions.

Let \mu _{2} be the population mean of students scores who go for mathematics tutions.

(2) Let n1 and n2 be the samples taken from the 2 populations and the degrees of freedom = n1 + n2 - 2

(3) Let \bar{x_{1}} and \bar{x_{2}} be the sample                means of n1 and n2 respectively.

(4) Let s1 and s2 be the sample standard deviations                  of n1 and n2 respectively.

(5) Let the confidence level be y%

(6) The Confidence interval

(\bar{x_{1}} -\bar{x_{2}}) -ME< \mu _{1}-\mu _{2}<(\bar{x_{1}} -\bar{x_{2}}) + ME

where

ME = t_{critical,(1-\frac{y}{100}), (n1+n2-2)}*\sqrt{\frac{s_{1}^{2}}{n1}+\frac{s_{2}^{2}}{n2}}

(7) The Hypothesis

H0: \mu _{1} = \mu _{2}     There is no                            difference in the 2 population means

Ha: \mu _{1} \neq \mu _{2}      There is a

difference between the 2 population Means

This is a 2 tailed test

(8) Let \alpha = y (If no value is mentioned we normally take a default \alpha = 0.05)

(9) The test Statistic is given by:

t_{observed} = \frac{\bar{x_{1}}-\bar{x_{2}}}{\sqrt{\frac{s_{1}^{2}}{n1}+\frac{s_{2}^{2}}{n2}}}

(10) If tobserved is > tcritical or if tobserved

is < -tcritical, then we Reject H0