Question

In 10000 meter race, runners complete over 25 laps or a 400-meter track. It is the longest track running event held. The men’s 10000 meter began at the 1912 Olympics. It was 1988 (over seventy years later) before the women’s 10000 meter was added to Olympic competition. The official world records in the 10,000 meters are held by Kenenisa Bekele with 26:17.53 minutes for men and Almaz Ayana from Ethiopia with 29:17.45 minutes for women.

In this project, we are going to take a look at the 2016 Rio Olympics men and women 10000 meter competition. All the times for the participant’s times are given to you in minutes. You should leave the times in minutes and round to the nearest thousandth when needed. Assume the populations are normally distributed.  

Country	Men's Name	Time in Minutes
KEN	Paul Kipngetich TANUI	27.09
ETH	Yigrem DEMELASH	27.11
UGA	Joshua Kiprui CHEPTEGEI	27.17
ERI	Zersenay TADESE	27.40
BRN	Abraham Naibei CHEROBEN	27.53
NZL	Zane ROBERTSON	27.56
USA	Leonard Essau KORIR	27.59
AUS	David MCNEILL	27.86
RSA	Stephen MOKOKA	27.91
BEL	Bashir ABDI	28.03
UGA	Moses Martin KURONG	28.06
ERI	Goitom KIFLE	28.27
BRN	El Hassan ELABBASSI	28.34
AUS	Ben ST LAWRENCE	28.77
JPN	Kota MURAYAMA	29.04
CAN	Mohammed AHMED	29.55

Country	Women's Name	Time in minutes
KEN	Vivian Jepkemoi CHERUIYOT	29.54
KEN	Alice Aprot NAWOWUNA	29.89
USA	Molly HUDDLE	30.22
ETH	Gelete BURKA	30.45
AUS	Eloise WELLINGS	31.25
SWE	Sarah LAHTI	31.47
NED	Susan KUIJKEN	31.54
GBR	Jessica ANDREWS	31.60
JPN	Yuka TAKASHIMA	31.61
JPN	Hanami SEKINE	31.74
CAN	Natasha WODAK	31.89
UZB	Sitora HAMIDOVA	31.96
POR	Carla Salome ROCHA	32.10
NED	Jip VASTENBURG	32.15
ITA	Veronica INGLESE	32.20
MEX	Brenda FLORES	32.65
GBR	Beth POTTER	33.07

1. Use the samples to find the mean completion time of the for the
   1. male runners.
   2. female runners.
2. Use the samples to find the sample standard deviation of the completion times for the
   1. male runners.
   2. female runners.
3. Find a point estimate for the mean completion time of the
   1. male runners.
   2. female runners.
4. Construct a 95% confidence interval for the population mean completion time of the
   1. male runners.
   2. female runners.
5. Interpret the results of number 4.

Answer 1

Assuming that the given data is normally distributed,

By definition of Sample Mean,

Sample mean = Sum of observations in the sample / Total No. of observations in the sample.

a. The mean completion time for male runners

  

= 27.96

b.

The mean completion time for male runners

  

= 31.49

a. By definition, the standard deviation for x_i, I = 1,2,...n can be computed using the formula:

The sample standard deviation of the completion times for the male runners

  

= 0.708

b.

The sample standard deviation of the completion times for the female runners

  

= 0.960

a.  The best point estimates for the mean completion time of Males and Females are nothing but their sample means respectively:

The point estimate for the mean completion time of Male runners

= 27.96

The point estimate for the mean completion time of Female runners

= 31.49

A 95% confidence interval for the population mean is constructed using the formula:

where, t_0.05,n-1 can be obtained from t tables for alpha = 0.05 for (n-1) degrees of freedom.

a. A 95% confidence interval for the mean completion time for male runners is obtained as:

From t table:

= (27.583, 28.337)

b. A 95% confidence interval for the mean completion time for male runners is obtained as:

= (30.996, 31.984)

For male runners,

If 100 such repeated samples of this size are obtained, for each of which the 95% CI is computed, 95 out of 100 intervals would contain the true population mean completion time of male runners.

For female runners,

If 100 such repeated samples of this size are obtained, for which the 95% CI is computed, 95 out of 100 intervals would contain the true population mean completion time of female runners.