Respuesta :
Answer:
a) [tex][-0.134,0.034][/tex]
b) We are uncertain
c) It will change significantly
Step-by-step explanation:
a) Since the variances are unknown, we use the t-test with 95% confidence interval, that is the significance level = 1-0.05 = 0.025.
Since we assume that the variances are equal, we use the pooled variance given as
[tex]s_p^2 = \frac{ (n_1 -1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}[/tex],
where [tex]n_1 = 40, n_2 = 30, s_1 = 0.16, s_2 = 0.19[/tex].
The mean difference [tex]\mu_1 - \mu_2 = 10.85 - 10.90 = -0.05[/tex].
The confidence interval is
[tex](\mu_1 - \mu_2) \pm t_{n_1+n_2-2,\alpha/2} \sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2}} = (-0.05) \pm t_{68,0.025} \sqrt{\frac{0.03}{40} + \frac{0.03}{30}}[/tex]
[tex]= -0.05\pm 1.995 \times 0.042 = -0.05 \pm 0.084 = [-0.134,0.034][/tex]
b) With 95% confidence, we can say that it is possible that the gaskets from shift 2 are, on average, wider than the gaskets from shift 1, because the mean difference extends to the negative interval or that the gaskets from shift 1 are wider, because the confidence interval extends to the positive interval.
c) Increasing the sample sizes results in a smaller margin of error, which gives us a narrower confidence interval, thus giving us a good idea of what the true mean difference is.
