Question

To show they should provide adequate protection against head injuries, bicycle helmets must be able to withstand government impact tests without cracking more than 10% of the time. A new helmet design has just been introduced. The new helmet is thought to be more stylish than older helmets, but consumer groups are worried that it may not pass government impact tests and will not provide adequate protection. 2000 of the new helmets are randomly sampled and tested. Of these sampled helmets, 210 of them cracked. Test the hypothesis that the new helmets will crack more than 10% time when enduring the impact tests. Use .05 as the level of significance. a) State the null and alternative hypotheses using proper notation. b) What test are you using? c) Give the values of the test statistic, and the p-value. d) Use your p-value to make your decision regarding the null hypothesis. e) What is your conclusion regarding the whether or not the new helmet cracks more than 10% of the time. Be sure to use the phrase "statistically significant."

Answer 1

a)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.1
Alternative Hypothesis, Ha: p > 0.1

b)

singleproportion z test

c)

pcap = 210/2000 = 0.105

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.105 - 0.1)/sqrt(0.1*(1-0.1)/2000)
z = 0.75

P-value Approach
P-value = 0.2266

d)

As P-value >= 0.05, fail to reject null hypothesis.

e)

There is not "statistically significant." evidence to conclude that new helmet cracks more than 10% of the time