Statistics and Probability Answers

1. A pharmaceutical company produces a new medicine and they claimed that it will reduce the migraine pain very fast with 𝟖𝟓% accuracy. Design a decision rule for the process with the significance 𝟎. 𝟎𝟏 by apply the medicine to 𝟏𝟓𝟎 people.

2. Let the class marks of a certain population table are 𝟏𝟕, 𝟐𝟐, 𝟐𝟕, 𝟑𝟐 & 𝟑𝟕 and the corresponding

frequencies are 𝟗, 𝟏𝟑, 𝟖, 𝟏𝟎 & 𝟏𝟓.

(i) Construct the original classes.

(ii) Draw the histogram.

(iii) Find the mode graphically.

3. Consider the following classes.

Class (43-47) (48-52) (53-57) (58-62) (63-67)

Frequency 9 8 12 6 15

(i) Sketch the histogram and derive frequency polygon from it.

(ii) Sketch the Pie chart.

(iii) Find the cumulative frequency polygon. Hence, locate the 𝑫_𝟕 and 𝑸_𝟏.

(iv) Evaluate 𝑸_𝟑, 𝑴𝒆, 𝑫_𝟕, & 𝑷_𝟖𝟕 from the cumulative frequency polygon.

1.

Consider pdf of a Normal variable 𝑿 is defined as 𝒇(𝒙) = (1/√98𝝅)^{e^-((x+11)^2/98))}

i) Find mgf of 𝑿.

ii) Evaluate 𝑷(−𝟑 > −𝑿 > 𝟏𝟑).

iii) Find value of 𝑪 such that 𝑷(|𝑿 + 𝟏𝟏| ≥ 𝑪) = 𝟎. 𝟎𝟔𝟏𝟒.

iv) Find −𝒁_{𝟎.𝟎𝟎𝟑𝟏} and convert to 𝑿.

2.

Let 𝒀_𝟏 < 𝒀_𝟐 < 𝒀_𝟑 <𝒀_𝟒 < 𝒀_𝟓 be the order statistics of five independent observations 𝑿_𝟏, 𝑿_𝟐, 𝑿_𝟑, 𝑿_𝟒 & 𝑿_𝟓 each from distribution with pdf 𝒇(𝒙) = 𝒆^-2x; 𝟎 ≤ 𝒙 < ∞.

(i) Find pdf of the sample median and it’s mean.

(ii) Determine 𝑷(𝒀𝟒 ≥ 𝟎. 𝟓). 

where, 𝑮_𝒓 (𝒚) = \sum∑(ⁿ_r ) [𝑭(𝒚)] ^𝒓 [𝟏 − 𝑭(𝒚)] ^𝒏−𝒓 ,

𝒈_𝒓 (𝒚) = (𝒏!/(𝒏−𝒓)!(𝒓−𝟏)!) [𝑭(𝒚)] ^𝒓−𝟏 [𝟏 − 𝑭(𝒚)] ^𝒏−𝒓𝒇(𝒚) 

3. Let 𝑿 equals weight of a soap. A random sample of size 𝟏𝟐 of 𝑿 yielded with weights 𝟓𝟏𝟑, 𝟒𝟗𝟑, 𝟒𝟗𝟖, 𝟓𝟎𝟑, 𝟒𝟗𝟓, 𝟓𝟏𝟏, 𝟓𝟎𝟓, 𝟒𝟗𝟕, 𝟒𝟖𝟖, 𝟒𝟗𝟑, 𝟓𝟏𝟏 & 𝟓𝟎𝟑

grams respectively. Now order them increasingly and find Median, 𝑫_𝟕 and 𝑷_𝟒𝟑. Is

there any mode exists? Determine the semi-range of given weights.

Consider the pdf of a Normal variable 𝑿 is defined as 𝒇(𝒙) = (1/√98𝝅)^{e^-((x+11)^2/98))}

i) Find the mgf of 𝑿.

ii) Evaluate 𝑷(−𝟑 > −𝑿 > 𝟏𝟑).

iii) Find the value of 𝑪 such that 𝑷(|𝑿 + 𝟏𝟏| ≥ 𝑪) = 𝟎. 𝟎𝟔𝟏𝟒.

iv) Find −𝒁_{𝟎.𝟎𝟎𝟑𝟏} and convert to 𝑿.

Consider the pdf of a Normal variable 𝑿 is defined as "f(x)= (1\/\u221a(98pi)^e^-((x+11)^2\/98)"

i) Find the mgf of 𝑿.

ii) Evaluate 𝑷(−𝟑 > −𝑿 > 𝟏𝟑).

iii) Find the value of 𝑪 such that 𝑷(|𝑿 + 𝟏𝟏| ≥ 𝑪) = 𝟎. 𝟎𝟔𝟏𝟒.

iv) Find −𝒁_{𝟎.𝟎𝟎𝟑𝟏} and convert to 𝑿.

a proportion of 0.4 are not satisfied with the service offered by the store. In a random sample of 400 customers, what is the probability that between 156 and 240 customers will be satisfied with the service offered by the store