Food intake and weight gain. If we increase our food intake, we generally gain w
ID: 3204154 • Letter: F
Question
Food intake and weight gain. If we increase our food intake, we generally gain weight. Nutrition scientists can calculate the amount of weight gain that would be associated with a given increase in calories. In one study, 16 nonobese adults, aged 25 to 36 years, were fed 1000 calories per day in excess of the calories needed to maintain a stable body weight. The subjects maintained this diet for 8 weeks, so they consumed a total of 56,000 extra calories.15 According to theory, 3500 extra calories will translate into a weight gain of 1 pound. Therefore, we expect each of these subjects to gain 56,000/3500 16 pounds (lb). Here are the weights before and after the 8-week period, expressed in kilograms (kg): WTGAIN Subject 1 2 3 4 5 6 7 8 Weightbefore 55.7 54.9 59.6 62.3 74.2 75.6 70.7 53.3 Weightafter 61.7 58.8 66.0 66.2 79.0 82.3 74.3 59.3 Subject 9 10 11 12 13 14 15 16 Weightbefore 73.3 63.4 68.1 73.7 91.7 55.9 61.7 57.8 Weightafter 79.1 66.0 73.4 76.9 93.1 63.0 68.2 60.3 For each subject, subtract the weight before from the weight after to determine the weight change. Find the mean and the standard deviation for the weight change. Calculate the standard error and the margin of error for 95% confidence. Report the 95% confidence interval for weight change in a sentence that explains the meaning of the 95%. Convert the mean weight gain in kilograms to mean weight gain in pounds. Because there are 2.2 kg per pound, multiply the value in kilograms by 2.2 to obtain pounds. Do the same for the standard deviation and the confidence interval. Test the null hypothesis that the mean weight gain is 16 lb. Be sure to specify the null and alternative hypothesis, the test statistic with degrees of freedom, and the P-value. What do you conclude? Write a short paragraph explaining your results.Explanation / Answer
> Weightbefore=c(55.7, 54.9, 59.6, 62.3, 74.2, 75.6, 70.7, 53.3,
+ 73.3, 63.4, 68.1, 73.7, 91.7, 55.9, 61.7, 57.8)
> Weightafter=c(61.7,58.8, 66.0,66.2, 79.0, 82.3, 74.3, 59.3,
+ 79.1, 66.0, 73.4, 76.9, 93.1, 63.0, 68.2, 60.3)
>
>
> # a)
> Weightdifference=Weightafter-Weightbefore
> Weightdifference
[1] 6.0 3.9 6.4 3.9 4.8 6.7 3.6 6.0 5.8 2.6 5.3 3.2 1.4 7.1 6.5 2.5
>
> # b)
> Mean=mean(Weightdifference)
> Mean
[1] 4.73125
> S_deviation=sd(Weightdifference)
> S_deviation
[1] 1.745745
>
> # c)
> n=length(Weightdifference)
>
> Std_error=S_deviation*sqrt(1/n)
> Std_error
[1] 0.4364362
> Margin_of_error=2.4898*Std_error # t-value for 0.025 level of significance at 15 df.
> Margin_of_error
[1] 1.086639
> ### 95% Confidence interval
>
> Lower_Limit=Mean-Margin_of_error
> Lower_Limit
[1] 3.644611
> Upper_Limit=Mean+Margin_of_error
> Upper_Limit
[1] 5.817889
>
> d) Change to kilogram
Error: unexpected ')' in "d)"
> Weightdifference_Kg=Weightdifference*2.2
>
> Mean_Kg=mean(Weightdifference_Kg)
> S_deviation_Kg=sd(Weightdifference_Kg)
> S_deviation_Kg
[1] 3.840639
>
> ### 95% Confidence interval
> Lower_Limit_Kg=Mean_Kg-2.4898*S_deviation_Kg*sqrt(1/n)
> Lower_Limit_Kg
[1] 8.018144
> Upper_Limit_Kg=Mean_Kg+2.4898*S_deviation_Kg*sqrt(1/n)
> Upper_Limit_Kg
[1] 12.79936
>
> ## e) # one sample t-test
> # Ho: mu=16 VS H1: mu=/16
>
> t.test(Weightdifference,mu=16)
One Sample t-test
data: Weightdifference
t = -25.82, df = 15, p-value = 7.582e-14
alternative hypothesis: true mean is not equal to 16
95 percent confidence interval:
3.801008 5.661492
sample estimates:
mean of x
4.73125
> # Conclusion: The estimated p-value is 0.0000 and less than 0.05 level of significance.
> # Hence, we does not accept the null hypothesis and conclude that mean is not equal to 16lb.
>
f) The R function is
Weightbefore=c(55.7, 54.9, 59.6, 62.3, 74.2, 75.6, 70.7, 53.3,
73.3, 63.4, 68.1, 73.7, 91.7, 55.9, 61.7, 57.8)
Weightafter=c(61.7,58.8, 66.0,66.2, 79.0, 82.3, 74.3, 59.3,
79.1, 66.0, 73.4, 76.9, 93.1, 63.0, 68.2, 60.3)
# a)
Weightdifference=Weightafter-Weightbefore
Weightdifference
# b)
Mean=mean(Weightdifference)
Mean
S_deviation=sd(Weightdifference)
S_deviation
# c)
n=length(Weightdifference)
Std_error=S_deviation*sqrt(1/n)
Std_error
Margin_of_error=2.4898*Std_error # t-value for 0.025 level of significance at 15 df.
Margin_of_error
### 95% Confidence interval
Lower_Limit=Mean-Margin_of_error
Lower_Limit
Upper_Limit=Mean+Margin_of_error
Upper_Limit
d) Change to kilogram
Weightdifference_Kg=Weightdifference*2.2
Mean_Kg=mean(Weightdifference_Kg)
S_deviation_Kg=sd(Weightdifference_Kg)
S_deviation_Kg
### 95% Confidence interval
Lower_Limit_Kg=Mean_Kg-2.4898*S_deviation_Kg*sqrt(1/n)
Lower_Limit_Kg
Upper_Limit_Kg=Mean_Kg+2.4898*S_deviation_Kg*sqrt(1/n)
Upper_Limit_Kg
## e) # one sample t-test
# Ho: mu=16 VS H1: mu=/16
t.test(Weightdifference,mu=16)