Internet company Gurgle is carrying out testing on the efficiency of its search engine. A sample of 45 searches have been carried out and the time taken to display the results has been recorded for each search. The mean search time for the sample was calculated as 0.1734 seconds. The standard deviation of the search times for the sample was calculated as 0.0187 seconds. The population standard deviation of search times is unknown. Gurgle The upper bound can be calculated using the following formula: show variables U = xtt x - = 0.1734 + 2.0154 x 0.0187 V 45 = 0.17901819... 0.1790 seconds Rounded as last step The lower bound can be calculated using the following formula: show variables = x+t x - Vo 0.0187 = 0.1734 + (2.0154) x V 45 0.16778181... 0.1678 seconds Rounded as last step b) Feedback [4 out of 4] a ) You are correct. b) You are correct. Discussion a ) A 95 % confidence interval for the mean is constructed by taking the sample mean as the center of the interval and defining the width of the interval using the margin of error. The margin of error is calculated using the sampling distribution of the mean. According to the Central Limit Theorem (CLT), the sampling distribution of the mean follows a normal distribution with mean u and standard deviation o/vn. In most situations the population standard deviation () is unknown and thus the standard deviation of the sampling distribution of the mean is unknown. To solve this issue, the population standard deviation is approximated using the sample standard deviation (s). When this approximation is made, the normal distribution cannot be used to construct the confidence interval because (x w)/(s/Vn) is not normally distributed. This expression actually follows the Student's t distribution which has a similar shape to the normal distribution, except it is thicker at the tails. That is, it has a larger standard deviation. Therefore, to construct a 95% confidence interval when the population standard deviation is unknown, o is approximated by s and the corresponding t value obtained from the Student's t distribution. b) The critical value corresponding to a 95% confidence interval can be obtained using the Student's t distribution table