Question #337551

A teacher conducted a study to know if blended leaming improves the students performances. A class of 25 students of Grade 11 was surveyed and found out that their mean score was 83 with a standard deviation of 3. A

study from other country revealed that = 80 with a standard deviation of 4. Test the hypothesis at 0.10 level of significance.


1
Expert's answer
2022-05-06T10:49:55-0400

Difference of two independent normal variables

Let XX have a normal distribution with mean μX\mu_X and variance σX2.\sigma_X^2.

Let YY have a normal distribution with mean μY\mu_Y and variance σY2.\sigma_Y^2.

If XX and YYare independent, then XYX-Ywill follow a normal distribution with mean μXμY\mu_X-\mu_Y and variance σX2+σY2.\sigma_X^2+\sigma_Y^2.

μXY=8380=3\mu_{X-Y}=83-80=3sXY=(3)2+(4)2=5s_{X-Y}=\sqrt{(3)^2+(4)^2}=5

The following null and alternative hypotheses need to be tested:

H0:μ=0H_0:\mu=0

H1:μ0H_1:\mu\not=0

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is α=0.01,\alpha=0.01,

df=n1=251=24df=n-1=25-1=24 degrees of freedom, and the critical value for a two-tailed test is tc=2.79694.t_c=2.79694.

The rejection region for this two-tailed test is R={t:t>2.79694}R=\{t:|t|>2.79694\}

The t-statistic is computed as follows:


t=μXYμsXY/n=305/250.12t=\dfrac{\mu_{X-Y}-\mu}{s_{X-Y}/\sqrt{n}}=\dfrac{3-0}{5/\sqrt{25}}\approx0.12

Since it is observed that t=0.12<2.79694=tc,|t|=0.12<2.79694=t_c, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value for two-tailed α=0.01,\alpha=0.01, df=24,t=0.12df=24, t=0.12 is p=0.001071,p=0.001071, and since p=0.905482>0.01=α,p=0.905482>0.01=\alpha, it is then concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean μ\mu is different than 0, at the α=0.01\alpha=0.01 significance level.

Therefore, there is not enough evidence to claim that blended leaming improves the students performances, at the α=0.01\alpha=0.01 significance level.


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