Question

In an article, measurements of the concentration of an anti-fungal gel, in ng per square centimeter of skin, were made one hour after application for 49 individuals. Following are the results. 132.44 76.73 258.46 177.46 73.01 130.62 235.63 107.54 75.95 70.37 88.76 104.00 19.07 174.30 82.87 68.73 41.47 120.44 136.52 82.46 67.04 96.92 93.26 72.92 138.15 82.43 245.41 104.68 82.53 122.59 147.12 129.82 54.83 65.82 75.24 135.52 132.21 85.63 135.79 65.98 349.71 77.84 89.19 102.94 166.11 168.76 155.20 44.35 202.51 The authors claim that these data are well-modeled by a lognormal distribution. Construct an appropriate probability plot, and use it to determine whether the data support this claim. Please upload your response/solutions using the controls below.

Answer 1

define

X: measurements of the concentration of an anti-fungal gel, in ng per square centimeter of skin, made one hour after application

it is claimed that the data are well-modeled by a lognormal distribution.

i.e. X ~ lognormal distribution (mu,sigma²)

which means

logX~ Normal (mu,sigma²)

define Y=log X

to check whether whether the data support this claim we plot normal probability plot of Y=log X

if the data (X) come from a lognormal distribution then the point Y= log X will fall approximately along a straight line in nornmal probability plot graph otherwise the plotted points (log X) will exhibit departaure from this straight line.

I have used MINITAB 16

type the data in MINITAB

input x and y=logx

go to

graph----Probability Plot---Single--OK---choose variable name as y and select in in the box---Distribution---Normal---OK----OK

here is the normal probability plot

the plotted points fall nearly on a straight line. so the data Y come from a normal distribution.

hence the data (X) come from a lognormal distribution.

so, the data (X) supports the claim.

[

we can cross check it by drawing histogram of Y . if a smooth curve drawn over histogram shows approximately shaped nature or symmetry ,then the data Y is approximately normal.

it is better to apply normal probability plot after drawing a histogram to guess the nature of the data.

check that plotted y shows symmetric nature (which is a property of normal distribution ) .

]

we can draw probability plot using EXCEL also.

for that

1. type x and y values in two columns

2. sort y in ascending order

3. find rank (i) of sorted y values

4. calculate f_i = (i-0.375) / (n+0.25) ,where n is the number of observations

5. find the z score values for f_i 's using the code " NORMSINV(f_i ).

6. plot( select Insert---Scater--choose first diagram) y values along the horizontal axis and corresponding z values along the vertical axis

in this way we get normal probability plot in excel

Visually it can be seen that the plotted points are approximately along a straight line. So the data Y come from a normal distribution. Hence the data X come from a lognormal distribution.

hence the data support the claim that the data are well-modeled by a lognormal distribution.

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