It has been hypothesized that allergies result from a lack of early childhood exposure to antigens. If this hypothesis were true, then we would expect allergies to be more common in very hygienic households with low levels of bacteria and other infectious agents. To test this theory, researchers at the University of Colorado sampled the houses of 61 children 9-24 months old and recorded two variables: (1) whether the child tested positive for allergies and (2) the concentration of bacterial endotoxin in the house dust (endotoxin units per ml, EU/ml)1. The following are the endotoxin levels at the homes of the 51 children tested negative for allergies. 708.23 911.60 976.81 1316.63 262.74 9772.08 370.76 229.16 2570.51 891.19 3163.20 1777.65 1288.57 436.23 2631.63 1173.52 911.67 7942.42 740.32 356.92 1175.48 1480.55 2754.61 575.62 573.89 468.26 1000.71 364.22 1025.26 1022.04 645.41 363.57 977.47 1022.75 1860.63 371.13 174.73 399.68 1479.77 2882.96 601.99 1697.32 2291.00 646.49 1176.27 1995.43 955.54 1480.05 456.71 1174.70 5494.22 (a) Make a boxplot for the endotoxin levels at the homes of the 51 children without allergy. Comment on whether it is appropriate to construct a t-confidence interval for the mean endotoxin level at the homes of children without allergy. (b) Make a boxplot of the log of the endotoxin levels at the homes of the 51 children without allergy.Comment on whether it is more appropriate to construct a t confidence interval for the mean of the log endotoxin level at the homes of children without allergy. (c) Construct a 95% t-confidence interval for the mean of the log endotoxin level at the homes of children without allergy.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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It has been hypothesized that allergies result from a lack of early childhood exposure to antigens. If this hypothesis were true, then we would expect allergies to be more common in very hygienic households with low levels of bacteria and other infectious agents. To test this theory, researchers at the University of Colorado sampled the houses of 61 children 9-24 months old and recorded two variables: (1) whether the child tested positive for allergies and (2) the concentration of bacterial endotoxin in the house dust (endotoxin units per ml, EU/ml)1. The following are the endotoxin levels at the homes of the 51 children tested negative for allergies.

708.23 911.60 976.81 1316.63 262.74 9772.08 370.76 229.16 2570.51
891.19 3163.20 1777.65 1288.57 436.23 2631.63 1173.52 911.67 7942.42 740.32 356.92 1175.48 1480.55 2754.61 575.62 573.89 468.26 1000.71 364.22 1025.26 1022.04 645.41 363.57 977.47 1022.75 1860.63 371.13 174.73 399.68 1479.77 2882.96 601.99 1697.32 2291.00 646.49 1176.27 1995.43 955.54 1480.05 456.71 1174.70 5494.22

(a) Make a boxplot for the endotoxin levels at the homes of the 51 children without allergy. Comment on whether it is appropriate to construct a t-confidence interval for the mean endotoxin level at the homes of children without allergy.

(b) Make a boxplot of the log of the endotoxin levels at the homes of the 51 children without allergy.Comment on whether it is more appropriate to construct a t confidence interval for the mean of the log endotoxin level at the homes of children without allergy.

(c) Construct a 95% t-confidence interval for the mean of the log endotoxin level at the homes of children without allergy.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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