## Answers

For the given problem,

Let denote the population mean, with unknown standard deviation. We need to compare this mean to a hypothesized value .

To test: Vs say,

The appropriate statistical test to test the above hypothesis would be a One sample t test, with test statistic given by,

where, M = Sample mean, s^{2} = Sample variation

1.Decrease in sample variance

From the test statistic, we find that the sample variance lies in the denominator of the test statistic and hence, is inversely proportional to t.Therefore,

Decrease in sample variance (s^{2}) increases the t statistic.

2. Decrease in obtained difference

From the test statistic, we find that the difference lies in the numerator of the test statistic and hence, is directly proportional to t.Therefore,

Decrease in obtained difference decreases the t statistic.

3. Increase in sample size n:

From the t statistic,

We find that the difference n lies in the numerator of the test statistic and hence, is directly proportional to t.Therefore,

Increase in sample size n increases the t statistic.

4.

Decrease in significance level alpha from 0.05 to 0.01.

Since, the observed / computed test statistic t has nothing to do with the significance level (it is the critical value of t that is obtained for a particular alpha, to which this observed t is compared to arrive at a decision of its significance), the t statistic remains the same.

5.

Given:

Cohen's d can be computed using the formula:

For a student who scored 6.1,

and

And

where, df for one sample t test = n - 1 = 36 - 1 = 35:

= 0.397

Hence, using the guidelines for interpreting the effect sizes, for cohen's d:

Cohen's d | Effect size |

0.2 | Small |

0.5 | Medium |

0.8 | Large |

The cohen's d obtained (d = -0.8) shows a large treatment effect.

r | Effect size |

0.1 | Small |

0.3 | Medium |

0.5 | Large |

Here, which suggests large effect.

Hence, using r^{2}, we may say that there is a large effect.

For n = 121, M = 6.1, s = 1.2:

Cohen's d can be computed as:

= -0.33

Interpreting cohen's d, we may say that there is a small to medium treatment effect.

And t = 3.64, for 121 - 1 = 120 df

= 0.099

Here, which suggests a medium effect

Interpreting r^{2}, we may say that there is a medium treatment effect.