Question

Two types of phone operating system are being tested to determine if there is a difference in the proportions of system failures (crashes). Fifteen out of a random sample of 150 phones with OS1 had system failures within the first eight hours of operation. Ten out of another random sample of 150 phones with OS2 had system failures within the first eight hours of operation. OS2 is believed to be more stable (have fewer crashes) than OS1.

A) State the null and alternative hypotheses. (out of the following)

H₀: p_OS1 = p_OS2
H_a: p_OS1 < p_OS2

H₀: p_OS1 = p_OS2
H_a: p_OS1 ≠ p_OS2    

H₀: p_OS1 = p_OS2
H_a: p_OS1 > p_OS2

H₀: p_OS1 ≥ p_OS2
H_a: p_OS1 < p_OS2

H₀: p_OS1 ≤ p_OS2
H_a: p_OS1 > p_OS2

B)What can you conclude about the two operating systems? (Use a significance level of 0.05.) (out of the following)

There is not sufficient evidence to reject the null hypothesis, so the data do not show that OS₂ has fewer system failures than OS₁.

There is not sufficient evidence to fail to reject the null hypothesis, so the data does show that OS₂ has fewer system failures than OS₁.    

There is sufficient evidence to reject the null hypothesis, so the data does show that OS₂ has fewer system failures than OS₁.

There is not sufficient evidence to fail to reject the null hypothesis, so the data do not show that OS₂ has fewer system failures than OS₁.

Answer 1

a)

H0: pOS1 ≤ pOS2
Ha: pOS1 > pOS2

b)

p1cap = X1/N1 = 15/150 = 0.1
p1cap = X2/N2 = 10/150 = 0.0667
pcap = (X1 + X2)/(N1 + N2) = (15+10)/(150+150) = 0.0833

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.1-0.0667)/sqrt(0.0833*(1-0.0833)*(1/150 + 1/150))
z = 1.04

P-value Approach
P-value = 0.1492
As P-value >= 0.05, fail to reject null hypothesis.

There is not sufficient evidence to reject the null hypothesis, so the data do not show that OS2 has fewer system failures than OS1.