Question

Do various occupational groups differ in their diets? A British study of this question compared 92 drivers and 69 conductors of London double-decker buses. The conductors' jobs require more physical activity. The article reporting the study gives the data as "Mean daily consumption ± (se)." Some of the study results appear below.

	Drivers	Conductors
Total calories	2825 ± 13	2847 ± 18
Alcohol (grams)	0.25 ± 0.09	0.37 ± 0.11

(a) Give x and s for each of the four sets of measurements. (Give answers accurate to 3 decimal places.)
Drivers Total Calories: x =  
s =  
Drivers Alcohol: x =  
s =  
Conductors Total Calories: x =  
s =  
Conductors Alcohol: x =  
s =  

(b) Is there significant evidence at the 5% level that conductors consume more calories per day than do drivers? Use the conservative two-sample t method to find the t-statistic, and the degrees of freedom. (Round your answer for t to three decimal places.)

t =
df =

Conclusion

Reject H₀.Do not reject H₀.    

(c) How significant is the observed difference in mean alcohol consumption? Use the conservative two-sample tmethod to obtain the t-statistic. (Round your answer to three decimal places.)
t =  Conclusion

Reject H₀.Do not reject H₀.    

(d) Give a 95% confidence interval for the mean daily alcohol consumption of London double-decker bus conductors. (Round your answers to three decimal places.)
(  ,  )

(e) Give a 99% confidence interval for the difference in mean daily alcohol consumption for drivers and conductors. (conductors minus drivers. Round your answers to three decimal places.)
(  ,  )

Answer 1

Solution:-

a)

Drivers Total Calories: x = 2825
s = 13
Drivers Alcohol: x = 0.25
s = 0.09

Conductors Total Calories: x = 2847

s = 18

Conductors Alcohol: x = 0.37

s = 0.11

b)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u₁> u ₂  

Alternative hypothesis: u ₁ < u ₂

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the mean difference between sample means is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]

SE = 2.556

DF = 92 + 69 - 2

D.F = 159

t = [ (x₁ - x₂) - d ] / SE

t = - 8.61

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is the size of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

The observed difference in sample means produced a t statistic of - 8.61 We use the t Distribution Calculator to find P(t < - 8.61) = less than 0.0001

Therefore, the P-value in this analysis is less than 0.0001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that conductors consume more calories per day than do drivers.

c)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u ₁ - u ₂ = 0
Alternative hypothesis: u ₁ - u ₂ 0

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]

SE = 0.01623

DF = 92 + 69 - 2

D.F = 159

t = [ (x₁ - x₂) - d ] / SE

t = - 7.394

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is the size of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 159 degrees of freedom is more extreme than -7.394; that is, less than -7.394 or greater than 7.394.

Thus, the P-value = less than 0.001

Interpret results. Since the P-value (0.001) is less than the significance level (0.05), we have to reject the null hypothesis.

Reject the null hypothesis.

d) 95% confidence interval for the mean daily alcohol consumption of London double-decker bus conductors is C.I = (0.3436, 0.3964).

C.I = 0.37 + 1.995 × 0.01324

C.I = 0.37 + 0.02642

C.I = (0.3436, 0.3964)