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Iron deficiency anemia is an important nutritional health problem in the U.S. A dietary assessment was performed on 51 boys 9-11-years old whose families were below the poverty level. The mean daily iron intake among these boys was found to be 12.50 mg with standard deviation 4.75 mg. Suppose the mean daily iron intake among a large population of 9-11-years old boys from all income strata is 14.44 mg. We want to test if the mean iron intake among the low-income group is different from that of the general population.
a. State the hypothesis that you can use to consider this question.
b. Carry out the hypothesis test using the critical-value method with an alpha= 0.05
c. Construct a 95% confidence interval estimate for the true mean of iron intake of the general population.

Respuesta :

Answer:

a) Null hypothesis:[tex]\mu = 14.44[/tex]    

Alternative hypothesis:[tex]\mu \neq 14.44[/tex]    

b) [tex]t=\frac{12.50-14.44}{\frac{4.75}{\sqrt{51}}}=-2.917[/tex]    

The degrees of freedom are given by:

[tex] df =n-1= 51-1=50[/tex]

Now we can calculate the p value taking in count the alternative hypothesis

[tex]p_v =2*P(t_{50}<-2.917)=0.0053[/tex]

Since the p value is lower than the significance [tex]\alpha=0.05[/tex] we have enough evidence to reject the null hypothesis and the true mean is significantly different from 14.44

c) [tex] 12.50 -2.01 \frac{4.75}{\sqrt{51}}= 11.163[/tex]

[tex] 12.50 +2.01 \frac{4.75}{\sqrt{51}}= 13.837[/tex]

Step-by-step explanation:

Information given

[tex]\bar X=12.50[/tex] represent the mean for the daily iron intake

[tex]s=4.75[/tex] represent the sample deviation

[tex]n=51[/tex] sample size    

[tex]\mu_o =14.44[/tex] represent the reference value  

[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.  

t would represent the statistic

[tex]p_v[/tex] represent the p value for the test

Part a

We want to test if the mean iron intake among the low-income group is different from that of the general population, the system of hypothesis would be:    

Null hypothesis:[tex]\mu = 14.44[/tex]    

Alternative hypothesis:[tex]\mu \neq 14.44[/tex]  

Part b  

Since we don't know the population deviation the statistic would be given by:

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)    

Replacing the info we got

[tex]t=\frac{12.50-14.44}{\frac{4.75}{\sqrt{51}}}=-2.917[/tex]    

The degrees of freedom are given by:

[tex] df =n-1= 51-1=50[/tex]

Now we can calculate the p value taking in count the alternative hypothesis

[tex]p_v =2*P(t_{50}<-2.917)=0.0053[/tex]

Since the p value is lower than the significance [tex]\alpha=0.05[/tex] we have enough evidence to reject the null hypothesis and the true mean is significantly different from 14.44

Part c

The confidence interval would be given by:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]

For the 95% confidence interval we can find the critical value in a t distribution with 50 degrees of freedom and we got:

[tex]t_{\alpha/2}= 2.01[/tex]

And replacing we got:

[tex] 12.50 -2.01 \frac{4.75}{\sqrt{51}}= 11.163[/tex]

[tex] 12.50 +2.01 \frac{4.75}{\sqrt{51}}= 13.837[/tex]