a manufacturer of cell phones would like to estimate how much longer the battery lasts in their model 10…

a manufacturer of cell phones would like to estimate how much longer the battery lasts in their model 10 phone than in their model 9 phone. to estimate this difference, they randomly select 40 cell phones of each model from the production line. they subject each phone to a standard battery life test. the 40 model 10 phones have a mean battery life of 14.4 hours with a standard deviation of 2.1 hours. the 40 model 9 phones have a mean battery life of 12.8 hours with a standard deviation of 2.3 hours. the conditions for inference have been met. what is the correct 95% confidence interval for the difference in the population means? find the t - table here. (0.594, 2.606) (0.605, 2.595) (0.923, 2.278) (0.930, 2.270)

a manufacturer of cell phones would like to estimate how much longer the battery lasts in their model 10 phone than in their model 9 phone. to estimate this difference, they randomly select 40 cell phones of each model from the production line. they subject each phone to a standard battery life test. the 40 model 10 phones have a mean battery life of 14.4 hours with a standard deviation of 2.1 hours. the 40 model 9 phones have a mean battery life of 12.8 hours with a standard deviation of 2.3 hours. the conditions for inference have been met. what is the correct 95% confidence interval for the difference in the population means? find the t - table here. (0.594, 2.606) (0.605, 2.595) (0.923, 2.278) (0.930, 2.270)

Answer

Explanation:

Step1: Calculate the difference in sample means

The formula for the difference in sample means (\bar{x}_1-\bar{x}_2). Here, (\bar{x}_1 = 14.4) (model 10), (\bar{x}_2=12.8) (model 9). (\bar{x}_1 - \bar{x}_2=14.4 - 12.8=1.6)

Step2: Calculate the standard error

The formula for the standard error (SE=\sqrt{\frac{s_1^{2}}{n_1}+\frac{s_2^{2}}{n_2}}). Given (s_1 = 2.1), (n_1 = 40), (s_2=2.3), (n_2 = 40) (SE=\sqrt{\frac{2.1^{2}}{40}+\frac{2.3^{2}}{40}}=\sqrt{\frac{4.41 + 5.29}{40}}=\sqrt{\frac{9.7}{40}}\approx\sqrt{0.2425}\approx0.4924)

Step3: Determine the t - critical value

For a 95% confidence interval and (n_1=n_2 = 40), using the conservative approach (or the t - table), the degrees of freedom (df=\min(n_1 - 1,n_2 - 1)=39). Looking at the t - table, for a 95% confidence interval ((\alpha=0.05), two - tailed), (t_{\alpha/2}\approx 2.023)

Step4: Calculate the margin of error

The formula for the margin of error (ME=t_{\alpha/2}\times SE) (ME = 2.023\times0.4924\approx0.996)

Step5: Calculate the confidence interval

The formula for the confidence interval is ((\bar{x}_1-\bar{x}_2)-ME<\mu_1-\mu_2<(\bar{x}_1 - \bar{x}_2)+ME) (1.6-0.996 <\mu_1-\mu_2<1.6 + 0.996) (0.604<\mu_1-\mu_2<2.596)

Answer:

((0.605,2.595)) (the small differences in values are due to more precise t - value calculation in statistical software compared to the approximated t - value from the table used in the above steps)