Reference no: EM13853763

Question 1: Simulating From The Sampling Distribution and Hypothesis Testing

 

A disease simulator can be found at the following link: https://www.learner.org/courses/envsci/interactives/disease/disease.html

 

The simulator models the spread of a disease across a population over the course of 100 days. We will be using this application to simulate an outbreak of Ebola in a high density region. According to research Ebola has a transmission rate of 15% and is fatal for 60% those infected. Assume that those infected with the Ebola virus will be contagious for 25 days. Set these to be the parameter values under the details tab of the application. Also set the population density to be High and the population mixing to Medium.

 

1. Run the simulation 50 times and record the proportions of people that die from each outbreak. Generate a histogram of your data and compute the four quantities that correspond to the histogram's shape characteristics.

 

Population: 900
Outbreak	Number of people that die for Ebola	% of deaths
1	504	56
2	515	57.2
3	515	57.2
4	518	57.6
5	519	57.7
6	522	58
7	523	58.1
8	524	58.2
9	526	58.4
10	528	58.7
11	528	58.7
12	528	58.7
13	528	58.7
14	529	58.8
15	529	58.8
16	532	59.1
17	533	59.2
18	533	59.2
19	534	59.3
20	536	59.6
21	536	59.6
22	537	59.7
23	539	59.9
24	539	59.9
25	541	60.1
26	542	60.2
27	542	60.2
28	543	60.3
29	543	60.3
30	544	60.4
31	545	60.6
32	545	60.6
33	546	60.7
34	546	60.7
35	548	60.9
36	550	61.1
37	550	61.1
38	551	61.2
39	553	61.4
40	554	61.6
41	559	62.1
42	559	62.1
43	559	62.1
44	562	62.4
45	566	62.9
46	566	62.9
47	566	62.9
48	567	63
49	571	63.4
50	578	64.2

 

Sturge's Formula (Class Intervals) = 1+3.3LOG10(50)

 

= 6.606601014

 

»7

 

Class Width = (Max-Min)/Class Interval

 

= (64.2-56)/7

 

»1.17

 

» 1

 

 

Descriptive Statistics
Mean	60.114
Standard Error	0.25704776
Median	60.15
Mode	58.7
Standard Deviation	1.817602143
Sample Variance	3.303677551
Kurtosis	-0.404838561
Skewness	0.140767785
Range	8.2
Minimum	56
Maximum	64.2
Sum	3005.7
Count	50

 

Empirical Rule
mean +/- 1 S.D.	58.29639786	61.93160214	32	64%
mean +/- 2 S.D.	56.47879571	63.74920429	48	96%
mean +/- 3 S.D.	54.66119357	65.56680643	50	100%

 

The histogram has a relatively bell shaped curve.The mean and median of the distribution of the proportion of deaths are approximately the same as the histogram depicts the shape of the distribution as approximately symmetric. With a standard deviation of 1.82, it means that data spread is close to the mean this corresponds to the histogram where most of the results are seen clustered around the mean. The skewness is 0.14 and kurtosis is -0.40 which both are very close to 0 indicates that it has a normal and mesokurtic distribution. Since the curve is approximately bell shaped, meaning that the proportion of deaths has approximately a normal distribution.There are also no outliers in the above histogram.

 

1. If you were able to observe an infinite number of samples (i.e. run the simulation an infinite number of times), what shape would you expect your histogram of the proportion of deaths to look like? Please frame your answer in terms of the four fundamental shape characteristics and make sure to provide your reasoning.

 

If an infinite number of samples were to be observed, the shape of the histogram of the proportion of deaths will become more symmetric and bell shaped. Since the sample data of 50 outbreaks indicated that the distribution of the proportion of deaths is approximately symmetric and bell shaped, for infinite number of samples this will stabilize to become perfectly symmetric and bell-shaped.

 

2. Some news agencies have reported that Ebola kills 65% of those infected. If this were true, what would be the sampling distribution of the estimator ˆp of the proportion of people that die from an Ebola outbreak?

 

p^: X/n=0.65=65%

 

3. A researcher from the World Health Organization informs you that the proportion reported in the news is too high, that instead the true proportion of people that would die from an Ebola outbreak is 60%. You need to decide who to believe. Outline your problem in terms of the hypothesis testing framework and provide a corresponding visual depiction. Make sure to correctly label all the items in your diagram and clearly state any assumptions you are making.

 

Step 1:

 

H₀: p = 0.60

 

H_A: p = 0.65

 

Step 2:

 

Use Z = (p^-p)/Ö(p^*q-hat) as testing a single population proportion p (uses standard normal)

 

Step 3:

 

Testing at a significance level of 5% (a = 0.05)

 

Step 4:

 

Reject H₀if z > z_0.05 = 1.645

 

Step 5:

 

4. In deciding between these two states of the world, you utilize a sample of 900 observations and set the probability of making a Type II error to be 5%. Compute the critical value associated with this probability and provide a corresponding visual depiction. Make sure to correctly label all the items in your diagram and clearly state any assumptions you are making.

 

The Type II error is (when H₀: p=0.60): fail to reject the null when it is false. Using the wrong proportion (60%) of people that would die from Ebola when 65% is true.

 

Critical Value: Z_0.05 = 1.645

 

Standardize: ?