To determine if the mean launch temperature for flights with o-ring damage is significantly less than for flights with no o-ring damage, we can perform a two-sample t-test with equal variances. Here are the steps:
Step 1: Calculate the sample means and standard deviations for each group:
For flights with o-ring damage:
Sample mean: (43 + 57 + 58 + 63 + 70 + 70 + 75) / 7 = 63.14
Sample standard deviation: 13.42
For flights with no o-ring damage:
Sample mean: (66 + 67 + 67 + 67 + 68 + 69 + 70 + 70 + 72 + 73 + 75 + 76 + 76 + 78 + 79 + 81) / 16 = 72.56
Sample standard deviation: 5.69
Step 2: Calculate the pooled standard deviation:
s_p = sqrt(((n1-1)*s1^2 + (n2-1)*s2^2) / (n1+n2-2))
where:
n1 = sample size of flights with o-ring damage = 7
n2 = sample size of flights with no o-ring damage = 16
s1 = sample standard deviation of flights with o-ring damage = 13.42
s2 = sample standard deviation of flights with no o-ring damage = 5.69
s_p = sqrt(((7-1)*13.42^2 + (16-1)*5.69^2) / (7+16-2)) = 9.88
Step 3: Calculate the t-test statistic:
t = (x1 - x2) / (s_p * sqrt(1/n1 + 1/n2))
where:
x1 = sample mean of flights with o-ring damage = 63.14
x2 = sample mean of flights with no o-ring damage = 72.56
s_p = pooled standard deviation = 9.88
n1 = sample size of flights with o-ring damage = 7
n2 = sample size of flights with no o-ring damage = 16
t = (63.14 - 72.56) / (9.88 * sqrt(1/7 + 1/16)) = -2.70
Step 4: Calculate the degrees of freedom:
df = n1 + n2 - 2 = 7 + 16 - 2 = 21
Step 5: Determine the critical value of t at 5% level of significance and the corresponding p-value:
At 5% level of significance and 21 degrees of freedom, the critical value of t is ±2.08 (from a t-distribution table or calculator).
The p-value for a two-tailed test with t = -2.70 and df = 21 is 0.013 (from a t-distribution table or calculator).
Step 6: Compare the t-test statistic with the critical value and the p-value with the level of significance:
Since the absolute value of the t-test statistic (-2.70) is greater than the critical value of t at 5% level of significance (2.08), we reject the null hypothesis and conclude that there is a significant difference in mean launch temperature between flights with o-ring damage and flights with no o-ring damage.
Moreover, the p-value (0.013) is less than the level of significance (0.05), providing further evidence to reject the null hypothesis.
Therefore, we can say that the mean launch temperature for flights with o-ring damage is significantly less.