8. A bathroom company counts the number of visitors to their shop in Stirling each Sunday. A sample of these results is shown. 44 55 32 39 43 26 34 (a) For these results,calculate: (i) the mean (ii) the standard deviation. The number of visitors to the company's shop in Aberdeen each Sunday was also recorded. The mean number of visitors was 49 and the standard deviation was 3.2. (b) Make two valid comments about the number of visitors each Sunday to the shops in Stirling and Aberdeen.

## Step1: <br />To calculate the mean, we first count the number of observations. In this case, there are 7 observations. <br /><br />## Step2: <br />Next, we add all the observations together: \(44 + 55 + 32 + 39 + 43 + 26 + 34 = 273\)<br /><br />### The formula for the mean is: <br />**\(\mu = \frac{\Sigma x}{N}\)**, where \(\Sigma x\) is the sum of all observations and \(N\) is the number of observations.<br /><br />## Step3: <br />Now, we divide the sum by the number of observations: \(273 \div 7 = 39\)<br /><br />## Step4: <br />To calculate the standard deviation, we first find the difference of each observation from the mean, square it, and then sum up all these squared differences.<br /><br />## Step5: <br />The differences are: \(44 - 39 = 5\), \(55 - 39 = 16\), \(32 - 39 = -7\), \(39 - 39 = 0\), \(43 - 39 = 4\), \(26 - 39 = -13\), \(34 - 39 = -5\)<br /><br />## Step6: <br />The squared differences are: \(5^2 = 25\), \(16^2 = 256\), \((-7)^2 = 49\), \(0^2 = 0\), \(4^2 = 16\), \((-13)^2 = 169\), \((-5)^2 = 25\)<br /><br />## Step7: <br />The sum of the squared differences is: \(25 + 256 + 49 + 0 + 16 + 169 + 25 = 540\)<br /><br />### The formula for the variance is: <br />**\(\sigma^2 = \frac{\Sigma (x - \mu)^2}{N}\)**, where \(\Sigma (x - \mu)^2\) is the sum of the squared differences from the mean and \(N\) is the number of observations.<br /><br />## Step8: <br />The variance is: \(540 \div 7 = 77.14\)<br /><br />### The formula for the standard deviation is: <br />**\(\sigma = \sqrt{\sigma^2}\)**, where \(\sigma^2\) is the variance.<br /><br />## Step9: <br />The standard deviation is: \(\sqrt{77.14} = 8.78\)<br /><br />## Step10: <br />Comparing the mean and standard deviation of the visitors to the shops in Stirling and Aberdeen, we can make the following comments:<br /><br />1. The mean number of visitors to the shop in Aberdeen is higher than in Stirling, indicating that the shop in Aberdeen has a higher average number of visitors.<br />2. The standard deviation of the number of visitors to the shop in Aberdeen is lower than in Stirling, indicating that the number of visitors in Aberdeen is more consistent, with less variability from week to week.