Question

The Bright Idea Lighting Company manufacturers light bulbs. They claim that over half of their light bulbs last for at least 400 hours. A random sample of 21 light bulbs produced the following lifetimes (in hours): [190, 225, 265, 288, 297, 303, 314, 327, 335, 368, 387, 392, 401, 411, 426, 435, 440, 441, 448, 452, 463].

Let θ0.5 denote the true median lifetime of Bright Idea light bulbs. Consider testing the hypotheses H0 : θ0.5 = 400 vs. Ha : θ0.5 < 400 at the α = 0.10 significance level.

Based on the (normal approximation to the) binomial test, is there sufficient evidence to reject H0?

Answer 1

[190.0, 225.0, 265.0, 288.0, 297.0, 303.0, 314.0, 327.0, 335.0, 368.0, 387.0, 392.0, 401.0, 411.0, 426.0, 435.0, 440.0, 441.0, 448.0, 452.0, 463.0]

Since we know that

Where n is the number of data points

Now

and n = 21

This implies that

  

The test hypothesis is

Now, the value of test static can be found out by following formula:

Since the sample size is n = 21, degrees of freedom on the t-test statistic are n-1 = 21-1 = 20 This implies that   

Since, the t distribution is symmetric about zero, so -t_{0.1,20}

Since we reject the null hypothesis in favor of the alternative hypothesis .