Question

II. The goal of this problem is to simulate the distribution of the sample mean. We will use the buit load the dataset and avoid some problems, copy and paste the following command in dataset 1ynx. To lynx as.numeric(lynx) Assume this vector represents the population. Le, the mean of this vector is our "true mean" (a) Draw a histogram of the population, find the "true" mean, and the true" variance. Does this data look normally distributed? (b) Draw a random sample of size 10 from the population, and report back the mean of your sample. (c) Repeat part (b) 1000000 times. You should have a vector of 1000000 means. Find the mean of that vector, and the standard deviation of that vector. Plot a histogram. (d) Draw a random sample of size 50 from the population, and report back the mean of your sample. (e) Repeat part (d) 1000000 times. You shpuld have a vector of 1000000 means. Did the mean and standard deviation increase, decrease, or stay approximatety the same from (c)? (f) Plot a histogram of the vector in (e) and describe the shape. What did you notice as the sample size increased?

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Answer 1

Dataset and histogram options (scipen = 999) df <as.numeric(lynx) hist(df) Histogram of df >寸 0 1000 2000 3000 4000 5000 6000 7000 df round (mean (lynx) , 4 ) mean= variance-round (var (lynx),4) The mean of the dataset is 1538.0175 and the variance is 2514900.92

Sample of size 10 sample_10 <- sample (df, 10) mean_sam_10 <- round (mean (sample_10),4) The mean of the sample of size 10 is 1214.5 Function to sample with size- 10 and find the mean and standard deviation sam size 10 n= 1000000 sampling mean_10<-c) for (i in 1:n) sam <- sample (df, sam size) mean sam <- mean (sam) sampling mean 10ij<-mean sam i=i+1 mean_samp_10 <-mean (sampling mean_10) sd_samp_10 <-sd (sampling mean_10) hist (sampling mean_10, main -"Histogram of the mean of the sample size 10")

Histogram of the mean of the sample size 10 1000 2000 3000 4000 sampling_mean_10 For 1000000 samples of size 10, the mean is 1537.3356002 and the standard deviation is 478.1567902 Sample of size 50 sample_50< sample (df,50) mean_sam_50 - round (mean (sample_50),4) The mean of the sample of size 50 is 1488.64 Function to sample with size-50 and find the mean and standard deviation

Function to sample with size- 50 and find the mean and standard deviation sam size = 50 n-1000000 sampling mean 50x-c0 for (i in 1:n) sam<sample (df, sam_size) mean_sam-mean (sam) sampling_mean_50 [i]<- mean_sam i=i+1 mean_samp 50mean (sampling_mean_50) sd samp_50 <- sd(sampling_mean_50) hist (sampling_mean_50, main -"Histogram of the mean of the sample size 50") Histogram of the mean of the sample size 50

Histogram of the mean of the sample size 50 2000 1000 1500 sampling_mean_50 For 1000000 samples of size 50, the mean is 1537.9892117 and the standard deviation is 168.2595617 As the sample size increases the sampling distribution tends more to a normal distribution