Question

Let's illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the scores of 10 students on an exam, shown in the table Student 4 10 Score 94 638145727265517559 The parameter of interest is the mean score μ in this population. The sample is an SRS of size n = 4 drawn from the population. The Simple Random Sample applet can be used to select simple random samples of four numbers between 1 and 10, corresponding to the students (a) Use the applet to generate 25 samples of size 4, calculate x for each, and construct a histogram of the 25 values. You are constructing the sampling distribution of x. One repetition of 25 sample means resulted in the histogram provided O 3 2 60 65 70 75 Sample Means (b) Based on the histogram, what would you estimate to be the chance of obtaining a simple random sample of four students with x 278? Choose the description that matches these results most closely 2 C0 60 65 70 75 Sample Means (b) Based on the histogram, what would you estimate to be the chance of obtaining a simple random sample of four students with i 2 787 Choose the description that matches these results most closely. O The chance of obtaining x 2 78 is almost certain. The chance of obtaining 2 78 is very likely. The chance of obtaining 2 78 is about fifty-fifty. The chance of obtaining 2 78 is very unlikely. c) Suppose you learn that students 1,3,5, and 7 are honors students. Would you regard their mean score to be "statistically significant"? OThe mean for these four students is 78. Based on the results in part (a), we might conclude that the mean score for OThe mean for these four students is 80.5. Based on the results in part (a), we might conclude that the mean score OThe mean for these four students is 78. Based on the results in part (a), we might conclude that the mean score for The mean for these four students is 80.5. Based on the results in part (a), we might conclude that the mean score the four honors students was unusually high (that is, this mean is statistically significant) for the four honors students was not at all unexpected (that is, this mean is not statistically significant) the four honors students was not at all unexpected (that is, this mean is not statistically significant) for the four honors students was unusually high (that is, this mean is statistically significant)

Answer 1

a)

Mean of sample size 4 寸 CN 50 55 60 65 70 75 80 Mean

b)

Ans: The chance of obtaining xbar>78 is very unlikely.

c) Ans: The mean for the four students is 78. Based on the results in part a), we might conclude that the mean score for the four honors students are unusually high (that is, this mean is statistically significant).

### R commmand

data=c(94,63,81,45,72,72,55,51,75,59)

Mean_boot=function(data, nsim)
{
BOOT=list()
Mean=list()

for(i in 1:nsim)
{
BOOT[[i]]=sample(data, size=4, replace = TRUE)
Mean[[i]]=mean(BOOT[[i]])
}
return(Mean)
}

Mean=Mean_boot(data=data, nsim=25)
Mean=unlist(Mean)
Mean

## a)
hist(Mean, main="Mean of sample size 4")

## b)
sort(Mean)