5) The human carcinogen dioxin can be produced by the burning of common trash. C

Question

5) The human carcinogen dioxin can be produced by the burning of common trash. Curiously it is hypothesized that trash from homes that recycle will produce more dioxins per kilogram of trash burned than homes that don't recycle. It is thought that a higher proportion of chlorine and metals in the recyclers' trash is the cause of higher levels of dioxin per kilogram. Below are the dioxin levels in picograms/kg of trash from 15 families that are avid recyclers. Dioxin (picograms/kg) 251 247 207 281 290 260 265 190 212 150 264 230 219 250 177 (a) Calculate the mean and standard deviation for these data. (b) Calculate the 95% confidence intervals for the mean and variance of these data. (c) If a well-run municipal incinerator produces 200 picograms/kg of trash burned, do you feel the recyclers' trash produces significantly more dioxin per kilogram?

Explanation / Answer

Part a

For given data,

Mean = 232.8666667

Standard deviation = 39.88531177

Part b

Formula for confidence interval for mean is given as below:

Confidence interval = Xbar -/+ t*S/sqrt(n)

We are given

Xbar = 232.8666667

S = 39.88531177

Sample size = n = 15

Confidence level = 95%

Degrees of freedom = n – 1 = 15 – 1 = 14

Critical t value = 2.1448

Confidence interval = Xbar -/+ t*S/sqrt(n)

Confidence interval = 232.8666667 -/+ 2.1448*39.88531177/sqrt(15)

Confidence interval = 232.8666667 -/+ 2.1448* 10.29834322

Confidence interval = 232.8666667 -/+ 22.0877

Lower limit = 232.8666667 - 22.0877 = 210.78

Upper limit = 232.8666667 + 22.0877 = 254.95

Confidence interval = (210.78, 254.95)

Now, we have to find confidence interval for variance.

Confidence interval is given as below:

(n – 1)*S2 / 2/2, n – 1 < 2 < (n – 1)*S2 / 21 -/2, n– 1

We are given

Sample size = n = 15

Degrees of freedom = n – 1 = 14

Confidence level = 95%

Lower chi square value = 5.6287

Upper chi square value = 26.1189

Confidence interval is given as below:

(15 – 1)* 39.8853^2 / 26.1189 < 2 <(15 – 1)* 39.8853^2 /5.6287

852.7036 < 2 < 3956.7958

Confidence interval = (852.7036, 3956.7958)

Part c

Here, we have to use one sample t test for population mean.

H0: µ = 200 versus Ha: µ > 200

This is an upper tailed test. (Right tailed / one tailed test)

Test statistic = t = (Xbar - µ) / [S/sqrt(n)]

We are given

Xbar = 232.8666667

S = 39.88531177

Sample size = n = 15

df = n – 1 = 14

Default level of significance = = 0.05

Test statistic = t = (232.8666667 – 200) / [39.88531177/sqrt(15)]

Test statistic = t = 3.1915

Upper critical value = 1.7613

P-value = 0.0033

= 0.05

P-value < = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that the recyclers’ trash produces significantly more dioxin per kilogram.

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