Question

Given that a light bulb will have a lifespan of three months averagely with ........ a standard deviation of one month. Let five of these following light bulbs indicate the lifespan (in months) ......... ......... 4.2, 2.4, 3.5, 1.9, 3.0 ......... Can we still claim that the light bulbs have a standard deviation of 1 month? Assuming that the light bulb lifetime follows a normal distribution. 

Answer 1

$H_0: \sigma= 1 \\ H_1: \sigma ≠1 \\ \sum^5_{i=1} (x_i-3)^2 = (4.2-3)^2 + (2.4-3)^2 +(3.5-3)^2 +(1.9-3)^2 +(3-3)^2= 3.26$

Test-statistic:

$χ^2= \frac{\sum^5_{i=1} (x_i-3)^2}{1^2}=3.26$

p-value $= 2P(χ^2 ≤3.26| χ^2 ~ χ^2_5) = 0.6801$

Use R code:

round (2*pchisq(3.26,5),4)

Since p-value > 0.05, we fail to reject H₀ at 5% level of significance. Hence there is sufficient evidence to conclude that the light bulbs have s standard deviation of 1 month.