Question

The built-in R dataset swiss gives Standardized fertility measure and socio-economic indicators for each of 47 French-speaking provinces of Switzerland at about 1888. The dataset is a data frame containing 6 columns (variables). The column Infant.Mortality represents the average number of live births who live less than 1 year over a 3-year period. We are interested in the Infant.Mortality column. We can convert the data in this colun to an ordinary vector x by making the assignment x <- swiss$Infant.Mortality. Then we can easily access the data. We can also get the data by entering the values 1 by 1. (You would be wise to not do the 1 by 1 entry.) The following is a screen print of the data values:

[1] 22.2, 22.2, 20.2, 20.3, 20.6, 26.6, 23.6, 24.9, 21.0, 24.4, 24.5, 16.5, 19.1, 22.7
[15] 18.7, 21.2, 20.0, 20.2, 10.8, 20.0, 18.0, 22.4, 16.7, 15.3, 21.0, 23.8, 18.0, 16.3
[29] 20.9, 22.5, 15.1, 19.8, 18.3, 19.4, 20.2, 17.8, 16.3, 18.1, 20.3, 20.5, 18.9,, 23.0
[43] 20.0, 19.5, 18.0, 18.2, 19.3

Assume these values are a random sample from a normal population with unknown mean μ and unknown standard deviation σ.
Let x be the vector created by the assignment x <- swiss$Infant.Mortality.

k) Using this data, create a 99% confidence interval for μ, noting that the sample size is large enough so we can use a normal distribution critical value zstar.

l) Using this data, create a 99% prediction interval for μ, noting that the sample size is large enough so we can use a normal distribution critical value zstar.

m) Using this data, we create a 1% level test of H₀: μ=21 versus the alternative H_a: μ < 21. We will reject H₀if z = ((x-21)/s)/√47 < zstar where s is the sample standard deviation. What is the value of zstar? (Calculate from normal distribution)

n) Continuing from part m, what is the value of z?  

o) Continuing from parts m and n, what is the p value of the test.

Answer 1

Calculating the given sample data

Sample size (n)=47

Sample mean

Sample Variance

Sample Standard deviation

k) Confidence level =99%

hence

Now Z critical value for is

since the Sample size is greater than 30 so it is large enough so we can use a normal distribution critical value z star

Z=2.576

Now margin of error

Confidence interval

l)

since the Sample size is greater than 30 so it is large enough so we can use a normal distribution critical value z star

prediction level =99%

hence

Now Z critical value for is

Z=2.576

Now margin of error

prediction interval

m)

Now given that

Now test statistic

Significance level =1%

Since it is a upper tail test so The Z star for is

Since

so we can say that we have enough evidence to reject the null hypothesis

n)

Value of Z= - 2.495

o) from the standard probability distribution table the probability value for the Z is greater than -2.495 is

p=(1-0.0062)=0.9938