Question

The World Bank collected data on the percentage of GDP that a country spends on health expenditures ("Health expenditure," 2013) and also the percentage of women receiving prenatal care ("Pregnant woman receiving," 2013). The data for the countries where this information is available for the year 2011 are in following table:

Data of Health Expenditure versus Prenatal Care

Health Expenditure (% of GDP)	Prenatal Care (%)
3.7	54.6
5.2	93.7
5.2	84.7
10.0	100.0
4.7	42.5
4.8	96.4
6.0	77.1
5.4	58.3
4.8	95.4
4.1	78.0
6.0	93.3
9.5	93.3
6.8	93.7
6.1	89.8

Test at the 5% level for a correlation between percentage spent on health expenditure and the percentage of women receiving prenatal care. Then compute a 95% prediction interval for the percentage of woman receiving prenatal care for a country that spends 5.0 % of GDP on health expenditure.

(i) Which of the following statements correctly define both the null hypothesis H_Oand the alternate hypothesis H_A ?

A.   H_O :  ρ = 0   H_A :  ρ < 0

B.     H_O :  ρ > 0   H_A :  ρ = 0

C.   H_O :  ρ = 0   H_A :  ρ > 0

D.     none of these answers are correct

(ii) Enter the level of significance α used for this test, and the degrees of freedomdf:

Enter level of significance in decimal form to nearest hundredth, followed by comma, followed by degrees of freedom value to nearest integer. Do not enter spaces.  

Examples of correctly entered answers:  0.01,4    0.02,11    0.05,13    0.10,46

(iii)   Use technology to determine correlation coefficient r between independent variable (percent GDP spent on healthcare) and dependent variable (percent women receiving prenatal care)

Enter in decimal form to nearest ten-thousandth with sign. Examples of correctly entered answers:  

-0.0001    +0.0020    -0.0500    +0.3000    +0.7115

(iv)  Calculate and enter test statistic

Enter value in decimal form rounded to nearest hundredth, with appropriate sign (no spaces). Examples of correctly entered answers:

–2.10      –0.07        +0.60        +1.09

(v) Using tables, calculator, or spreadsheet: Determine and enter p-value corresponding to test statistic.

Enter value in decimal form rounded to nearest thousandth. Examples of correctly entered answers:

0.000     0.001     0.030     0.600      0.814 1.000

(vi) Comparing p-value and α value, which is the correct decision to make for this hypothesis test?  

A. Reject H_o

B. Fail to reject H_o

C. Accept H_o

D. Accept H_A

Enter letter corresponding to correct answer.

(vii) Select the statement that most correctly interprets the result of this test:

A. The result is statistically significant at .05 level of significance. Evidence supports the claim that there is a correlation between percent GDP spent on healthcare and percentage of women receiving prenatal care.

B. The result is statistically significant at .05 level of significance. There is not enough evidence to show that there is a correlation between percent GDP spent on healthcare and percentage of women receiving prenatal care.

C. The result is not statistically significant at .05 level of significance. Evidence supports the claim that there is a correlation between percent GDP spent on healthcare and percentage of women receiving prenatal care.

D. The result is not statistically significant at .05 level of significance.   There is not enough evidence to show that there is a correlation between percent GDP spent on healthcare and percentage of women receiving prenatal care.

Enter letter corresponding to most correct answer

(viii) Determine standard error of the estimate s_e:

Enter answer to nearest hundredth

(ix) Compute a 95% prediction interval for the percentage of woman receiving prenatal care for a country that spends 5.0 % of GDP on health expenditure. Do this by:

• finding t_c(two-tailed t-score using level of significance and df value found in part (ii))  https://www.danielsoper.com/statcalc/calculator.aspx?id=10

• find "y^" output value corresponding to x_o = 5.0 using line-of-best-fit regression equation y^ =mx_o + b

• determining bound E using the following formula: (s_e , n , sample mean for x , and sample variance for x all can be found using http://vassarstats.net/index.html )

E=(tc) (se) (1+1n+( xo−x )2(sx2)(n−1)‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√ )E=tc se 1+1n+ xo-x 2sx2n-1

• subtract, then add bound E to y^ value to create prediction interval.

Determine which of the following 95% prediction intervals is most correct:

A. 39.42% < y < 116.41%

B. 23.29% < y < 132.55%

C. 47.65% < y < 108.19%

D. 40.89% < y < 114.94%

Enter letter corresponding to most correct answer

(x) Which of the following statements correctly interprets the significance of the prediction interval?

A. We estimate with 95% confidence that the interval 39.42% < y < 116.41% contains the true percentage of women receiving prenatal care for a 5.0% of GDP expenditure on healthcare.

B. We estimate with 95% confidence that the interval 47.65% < y < 108.19% contains the true percentage of women receiving prenatal care for a 5.0% of GDP expenditure on healthcare.

C. We estimate with 95% confidence that the interval 40.89% < y < 114.94% contains the true percentage of women receiving prenatal care for a 5.0% of GDP expenditure on healthcare.

D. We estimate with 95% confidence that the interval 23.29% < y < 132.55% contains the true percentage of women receiving prenatal care for a 5.0% of GDP expenditure on healthcare.

Answer 1

Let be the true value of correlation coefficient between percentage spent on health expenditure and the percentage of women receiving prenatal care.

We want to test at the 5% level for a correlation between percentage spent on health expenditure and the percentage of women receiving prenatal care. We say that there is correlation between 2 variables if the true value is not equal to zero, as the correlation may be any value between -1 to +1

The hypotheses are

i) ans: D. None of these answers are correct

(ii) Enter the level of significance α used for this test, and the degrees of freedomdf:

We want to test at the 5% level. Hence the level of significance α used for this test is 0.05

There are n=14 observations in the sample. The degrees of freedom is n=2=14-2=12

ans: 0.05,12

(iii) Use technology to determine correlation coefficient r between independent variable (percent GDP spent on healthcare) and dependent variable (percent women receiving prenatal care)

Enter the following in Excel to get the correlation coefficient

get this

The sample correlation r=0.4962

ans: +0.4962

(iv) Calculate and enter test statistic

The test statistics is

ans: +1.98

(v) Using tables, calculator, or spreadsheet: Determine and enter p-value corresponding to test statistic.
This is a 2 tailed test (The alternative hypothesis has "not equal to")

Enter the following in Excel

=T.DIST.2T(1.98,12)

and get

p-value=0.071

ans: 0.071

(vi) Comparing p-value and α value, which is the correct decision to make for this hypothesis test?  

We will reject the null hypothesis if the p-value is less than the α value.

Here, the p-value is 0.071 and it is less than the α value = 0.05.

Here we do not reject the null hypothesis.

ans: B. Fail to reject Ho

(vii) Select the statement that most correctly interprets the result of this test:
ans: D. The result is not statistically significant at .05 level of significance.   There is not enough evidence to show that there is a correlation between percent GDP spent on healthcare and percentage of women receiving prenatal care.

(viii) Determine standard error of the estimate s_e:

Let  Y=the percentage of woman receiving prenatal care for a country

X= GDP on health expenditure

the regression line that we want to estimate is

Using Excel to get this bets fitting line

Setup using data--->data analysis--->regression

get this

the estimated regression line is

ans: The standard error of the estimate s_e is 16.28

(ix) Compute a 95% prediction interval for the percentage of woman receiving prenatal care for a country that spends 5.0 % of GDP on health expenditure.

Given we get the predicted value of y (the percentage of woman receiving prenatal care ) using

finding tc(two-tailed t-score using level of significance and df value found in part ii)

level of significance is 0.05, df=12

get this

We also get the sample mean of x using (either manually or using Excel function =AVERAGE(A2:A15) )

Calculate the sample standard deviation of X using (either manually or using Excel function =STDEV.S(A2:A15) )

We then calculate the margin of error E using

Finally we get the prediction interval using

ans: D. 40.89% < y < 114.94%

(x) Which of the following statements correctly interprets the significance of the prediction interval?
ans: C. We estimate with 95% confidence that the interval 40.89% < y < 114.94% contains the true percentage of women receiving prenatal care for a 5.0% of GDP expenditure on healthcare.