## Assignment Instructions

1. Use descriptive statistics to summarize the proportion experiencing headache (grouped by the history of previous concussion and total), the proportion experiencing any symptoms (grouped by the history of previous concussion and total), and the mean number of symptoms experienced (grouped by sex and total). Discuss any relevant findings.

2. Develop 95% confidence intervals for (a) the proportion of all student-athletes experiencing headaches, and (b) the mean number of symptoms experienced by student-athletes participating in cheerleading. Comment on whether the appropriate assumptions are met for these methods.

3. Conduct a hypothesis test at an α = 0.05 significance level to determine if there is a difference in the mean number of symptoms experienced between male student-athletes and the mean number of symptoms experienced by female student-athletes. Include your hypotheses, test statistic, P-value, and conclusion. Comment on whether the appropriate assumptions are met for this test.

4. Conduct a hypothesis test at an α = 0.05 significance level to determine if the proportion of student-athletes experiencing headaches is different from the proportion indicated in the research. Include your hypotheses, test statistic, P-value, and conclusion. Comment on whether the appropriate assumptions are met for this test.

5. Given your results from the previous questions, what recommendations would you make to the state high school athletic association?

## Assignment Solutions

Below are questions that would help provide insight into the findings of this research. Please answer each of these and provide interpretations of the results.

1. Use descriptive statistics to summarize the proportion experiencing headache (grouped by the history of previous concussion and total), the proportion experiencing any symptoms (grouped by the history of previous concussion and total), and the mean number of symptoms experienced (grouped by sex and total). Discuss any relevant findings.

**Number Headache**

Concussion History | Yes | NO |

NO | 33 | 15 |

Yes | 76 | 30 |

**Proportion Headache**

Concussion History | Yes | No |

No | 0.6875 | 0.3125 |

Yes | 0.716981 | 0.283019 |

Of those who have experienced concussion before, 71.698% of those have had headaches, compared to 68.75% of those who haven’t experienced concussion before.

Number of Symptoms | Female | Male |

Mean | 6 | |

Variance | 34.85714286 | 11.01807018 |

Observation | 78 | 76 |

The average number of symptoms experienced by male student-athletes is 2.40789 compared to 6 for female student-athletes.

2. Develop 95% confidence intervals for (a) the proportion of all student-athletes experiencing headaches, and (b) the mean number of symptoms experienced by student-athletes participating in cheerleading. Comment on whether the appropriate assumptions are met for these methods.

(a)

The 95% confidence interval for the proportion of all student-athletes experiencing headaches is (0.22038, 0.364036)

(b)

The 95% confidence interval the mean number of symptoms experienced by student-athletes participating in cheerleading is (4.815597, 10.44756).

3. Conduct a hypothesis test at an α = 0.05 significance level to determine if there is a difference in the mean number of symptoms experienced between male student-athletes and the mean number of symptoms experienced by female student-athletes. Include your hypotheses, test statistic, P-value, and conclusion. Comment on whether the appropriate assumptions are met for this test.

Null hypothesis: Average number of symptoms experienced between male student-athletes and the mean number of symptoms experienced by female student-athletes are equal

Alternate hypothesis: Average number of symptoms experienced between male student-athletes and the mean number of symptoms experienced by female student-athletes aren’t equal

**T-test: Two-Sample Assuming Equal Variances**

Female | Male | |

Mean | 6 | 2.407894737 |

Varience | 34.85714286 | 11.01807018 |

Observations | 78 | 76 |

Pooled Variance | 23.09444252 | |

Hypothesized Mean Difference | 0 | |

df | 152 | |

T-test | 4.637554754 | |

P(T,=t) one-tail | 3.77241E-06 | |

t Critical one tail | 1.654940175 | |

t Critical one tail | 1.654940175 | |

t Critical one tail | 1.654940175 |

The test statistic is equal to 4.6375, df = 152, p-value = 7.5448*10-6< 0.05. Therefore, at 0.05 significance level, we reject the null hypothesis and conclude that the average number of symptoms experienced between male student-athletes and the mean number of symptoms experienced by female student-athletes aren’t equal

4. Conduct a hypothesis test at an α = 0.05 significance level to determine if the proportion of student-athletes experiencing headaches is different from the proportion indicated in the research. Include your hypotheses, test statistic, P-value, and conclusion. Comment on whether the appropriate assumptions are met for this test.

The research suggests that the proportion of student-athletes experiencing headaches would be close to 30.5%.

Null hypothesis: the proportion of all student-athletes experiencing headaches is 30.5%

Alternate hypothesis: the proportion of all student-athletes experiencing headaches isn’t 30.5%

We will use the one-sample Z-test to test the hypothesis.

Headache | |

Yes | 45 |

Total | 154 |

Proportion | 0.292208 |

95% Confidence Interval | |

Lower | |

Upper | 0.364036 |

Hypothesized Proportion | 0.305 |

Z-Statistics | -0.3448 |

P-Value | 0.730247 |

The value of the Z-statistic is -0.3448, p-value = 0.73 > 0.05. Therefore, at a 0.05 significance level, we fail to reject the null hypothesis and conclude that the proportion of all student-athletes experiencing headaches is 30.5%

5. Given your results from the previous questions, what recommendations would you make to the state high school athletic association?

Better rules for concussion guidelines, so that those with previous concussions have recovered well enough to participate further in the competitions.