Question

2. (25pts) Two different brands of latex (water-based) paint are being considered for use in a large construction project. To choose between the brands, one of the key factors is the time it takes the paint to dry. Engineers sampled 22 specimens of each brand and measured the drying times (in hours) of each specimen. The data collected are given in Table 1 below Specimen Brand A Brand B Specin Brand A Brand B 3.5 2.7 3.9 4.2 4.6 2.7 3.3 4.7 3.9 4.5 つ·つ 5.2 4.0 6.2 13 14 5.4 4.8 4.9 6.6 4.3 4.9 3.4 3.3 4.2 5.3 3.7 3.0 4.0 2.8 16 5.3 4.3 6.0 5.3 3.7 6 18 19 20 21 4.2 2.9 4.4 9 10 3.9 5.2 Table 1: Latex paint data. Drying times (measured i hours) for two brands of paint a) Treat these samples as independent samples, one taken from the Brand A population and one taken from the Brand B population. At the α-0.05 significance level, perform a hypothesis test to determine whether or not the mean drying times for the two brands of paints are different. To gain full credit, you should provide the following: i Identify the parameters of interest, and their meanings in the context of ii State and check modeling assumptions. Hint: don't forget to consider iii State the null and alternative hypotheses, the test statistic, the critical this problem ual variances vs. unequal variances eq region, and the p-value. iv Draw a conclusion based on your test statistic and the critical region. Alsıo use the p-value to draw a conclusion. Do these decisions correspond? If not go back and check your work v Interpret your conclusion in the context of this problemm. (b) Calculate and interpret (in the context of this problem) a 95 percent confidence interval for the difference in the population means.

Answer 1

Solution

Let

X = drying time (in hours) of Brand A paint.

Y = drying time (in hours) of Brand B paint.

Then, X ~ N(µ₁, σ₁²) and Y ~ N(µ₂, σ₂²), where σ₁² = σ₂² = σ², say and σ² is unknown.

Part (a)

i) Parameters of interest: (µ₁, µ₂) and (σ₁, σ₂) Answer 1

   Meaning of parameters of interest: (µ₁, µ₂) are the two population means and (σ₁, σ₂) are the two

   population standard deviations. Answer 2

ii) Modelling Assumptions: σ₁² = σ₂² = σ², say and σ² is unknown. Answer 3

    [Assumptions confirmed . Details at the end]

iii) Hypotheses:

Null: H₀: µ1 = µ2 Vs Alternative: H_A: µ1 ≠ µ2 Answer 4

Test Statistic:

t = (Xbar - Ybar)/{s√(2/n)} Answer 5

where

s² = (s₁² + s₂²)/2;

Xbar and Ybar are sample averages and s₁,s₂ are sample standard deviations based on n observations each on X and Y.

tcal = - 4.7155 Answer 5

Calculations

Summary of Excel calculations is given below:

n	22
Xbar	3.85
Ybar	4.959091
s1	0.799242
s2	0.766972
s^2	0.613517
s	0.783273
|tcal|	4.715485
α	0.05
tcrit	2.018082
p-value	2.67E-05

Critical Value = upper (α/2)% point of t_{2n – 2}

= upper 2.5% point of t₄₂

= 2.018 Answer 6

[obtained using Excel Function Statistical TINV]

p-value = P(t_{2n - 2} > | tcal |)

= P(|t₄₂| > 4.7155)

= 2.67E-05 Answer 7

[obtained using Excel Function Statistical TDIST]

Conclusion:

Critical value approach: Since | tcal | > tcrit, H₀ is rejected. Answer 8

p-value approach: Since p-value < α, H₀ is rejected. Answer 9

Comparison: The conclusions arrived at with both approaches are identical. Answer 10

Interpretation

There is sufficient evidence to suggest that the

mean drying time are different for the two brands of paints. Answer 11

Part (b)

95% confidence interval for the difference in the two population means is:

(Xbar – Ybar) ± {(t_{2n – 2, 0.025})(s)√(2/n)}

= [- 1.6151, - 0.6122] [Brand A – Brand ] Answer 12

DONE