**Testing Hypotheses for Means**

**Instructions:-**

## Testing Hypotheses for Means

**Scenario 1**

A research was carried out to determine whether the African Citizens’ perceptions about current levels of democracy on a scale of 1-10 is significantly different from 6 that leaders and development experts would like it to be. The null hypothesis was that there is no significant difference between the citizens’ perceptions and what is expected by the leaders and development experts. The alternative hypothesis is that there is a significant difference between the sample mean and the expected mean. A one-sample t-test analysis was carried out using IBM SPSS (Wagner, 2016).

The p-value is <0.05, therefore, there is enough evidence to reject the null hypothesis. It was concluded that the mean perceptions of African Citizens about current levels of democracy is significantly different from 6. The sample mean is 5.52, which is separated from the hypothesized mean of 6 by -35.924 standard errors (0.013). There is a 95% confidence that the mean difference between the current levels of democracy will lie between -0.50 and -0.45. The results suggest that the citizens perceive that the level of democracy in Africa is still lower than desired, which is a meaningful conclusion. Leaders need to engage in practices and enact policies that can ensure that the citizens feel that democracy is being upheld.

**Scenario 2**

Previous research and hypothesis tests have determined that perceptions about the current levels of democracy are significantly lower than 6. There were recent social change movements in North Africa. As a result, the researcher was interested in determining whether any statistical difference exists in the perceptions about democracy between North Africa and Southern Africa. The null hypothesis is that there is no difference between perceptions in North Africa and Southern Africa. The alternative hypothesis is that the perceptions of levels of democracy in the two regions are different. The researcher used an independent sample t-test to carry out the analysis in IBM SPSS (Laureate Education, 2016).

The independent t-test statistic without assuming equal variances is 18.510 and the mean difference is 0.879 which lies between 0.786 and 0.972 at the 95% confidence interval. Since the p-value is <0.05, there is a significant difference between perceptions of democracy between Southern Africa and North Africa. There is enough evidence at the 95% level of confidence to reject the null hypothesis and conclude that the perceptions about democracy levels in North Africa is different from that in Southern Africa. The social change movements in North Africa were effective in changing the people’s perceptions about levels of democracy. It is possible that such movements can also be useful in improving views of democracy in Southern Africa.

**Scenario 3**

To determine whether students change their perceptions about mathematical utility as they move from freshman to senior year, students were asked similar series of questions on how they perceive they will utilize mathematics in their future when they were in their freshman year and later on when they were in their senior year. The data collected was then analysed using paired samples t-test in IBM SPSS (Klingenberg, 2016). The null hypothesis was that there is no change in perceptions about mathematical utility between freshman and senior years. The alternative hypothesis is that the perceptions about mathematical utility change as a student moves from freshman year to senior year.

The mean perception in freshman year
is -0.0096 while that for senior year is 0.0059, which appears to be a large
enough difference. However, a statistical test is necessary to check whether
the change is significant or merely due to chance. The mean difference is
-0.016 with a 95% chance of the actual difference falling between -0.03404 and
0.00293. The test statistic is -1.649 with a p-value of 0.099. Since the
p-value > 0.05, there is not enough evidence to reject the null hypothesis.
It can therefore be concluded that the students’ perceptions of mathematical
utility do not change as they move from freshman year to senior year. However,
the conclusion would not be meaningful. As students get more exposed to more
concepts of mathematics through continued interaction with it, it is highly
likely that they get more insight into how useful mathematics can be to them in
future thereby increasing their perceptions as they move from freshman years to
senior years.

**References**

Klingenberg, B. (2016) Inference for comparing two population means. Retrieved from https://istats.shinyapps.io/2sample_mean/

Laureate Education. (2016). The t test for related samples [Video file]. Baltimore, MD.

Wagner, W.E. (2016). Using IBM® SPSS® statistics for research
methods and social science statistics (6^{th} ed.) Thousand Oaks, CA:
Sage Publications.

Appendix

One Sample T-Test

One-Sample Statistics
| ||||

N | Mean | Std. Deviation | Std. Error Mean | |

Q46a. Level of democracy: today | 46940 | 5.52 | 2.883 | .013 |

One-Sample Test
| ||||||

Test Value = 6 | ||||||

t | df | Sig. (2-tailed) | Mean Difference | 95% Confidence Interval of the Difference | ||

Lower | Upper | |||||

Q46a. Level of democracy: today | -35.924 | 46939 | .000 | -.478 | -.50 | -.45 |

Independent Sample T-Test

Group Statistics
| |||||

Country by region | N | Mean | Std. Deviation | Std. Error Mean | |

Q46a. Level of democracy: today | Southern Africa | 15979 | 5.78 | 2.795 | .022 |

North Africa | 5418 | 4.90 | 3.092 | .042 |

Independent Samples Test
| ||||||||||

Levene’s Test for Equality of Variances | t-test for Equality of Means | |||||||||

F | Sig. | t | df | Sig. (2-tailed) | Mean Difference | Std. Error Difference | 95% Confidence Interval of the Difference | |||

Lower | Upper | |||||||||

Q46a. Level of democracy: today | Equal variances assumed | 130.649 | .000 | 19.453 | 21395 | .000 | .879 | .045 | .790 | .967 |

Equal variances not assumed | 18.510 | 8610.815 | .000 | .879 | .047 | .786 | .972 |

Paired Sample T-Test

Paired Samples Statistics
| |||||

Mean | N | Std. Deviation | Std. Error Mean | ||

Pair 1 | T1 Scale of student’s mathematics utility | -.0096 | 16021 | .99040 | .00782 |

T2 Scale of student’s mathematics utility | .0059 | 16021 | 1.00682 | .00795 |

Paired Samples Test
| |||||||||

Paired Differences | t | df | Sig. (2-tailed) | ||||||

Mean | Std. Deviation | Std. Error Mean | 95% Confidence Interval of the Difference | ||||||

Lower | Upper | ||||||||

Pair 1 | T1 Scale of student’s mathematics utility – T2 Scale of student’s mathematics utility | -.01556 | 1.19384 | .00943 | -.03404 | .00293 | -1.649 | 16020 | .099 |

SPSS Syntax

DATASET ACTIVATE DataSet4.

T-TEST

/TESTVAL=6

/MISSING=ANALYSIS

/VARIABLES=Q46A

/CRITERIA=CI(.95).

T-TEST GROUPS=COUNTRY.BY.REGION(3 4)

/MISSING=ANALYSIS

/VARIABLES=Q46A

/CRITERIA=CI(.95).

DATASET ACTIVATE DataSet5.

T-TEST PAIRS=X1MTHUTI WITH X2MTHUTI (PAIRED)

/CRITERIA=CI(.9500)

/MISSING=ANALYSIS.