Question

an app estimates phone usage by counting the number of times a phone screen is unlocked during the course of a day. a sample of 25 users is shown. complete123 65 119 145 114125 114 91 113 12588 141 105 116 121186 136 65 128 10797 126 101 90 10a. what is the mean and standard deviation?the mean is 110.04(round to the nearest hundredth as needed )the standard deviation is (round to the nearest hundredth as needed )

Explanation:

Step1: Recall the formula for sample standard deviation

The formula for the sample standard deviation \( s \) is \( s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}} \), where \( x_{i} \) are the individual data points, \( \bar{x} \) is the sample mean, and \( n \) is the sample size. We know \( n=25 \) and \( \bar{x} = 110.04 \).

Step2: Calculate \( (x_{i}-\bar{x})^{2} \) for each data point

First, list all the data points: \( 123, 65, 119, 145, 114, 125, 114, 91, 113, 125, 88, 141, 105, 116, 121, 186, 136, 65, 128, 107, 97, 126, 101, 90, 10 \)

For each \( x_{i} \):

• \( (123 - 110.04)^{2}=(12.96)^{2}=167.9616 \)
• \( (65 - 110.04)^{2}=(- 45.04)^{2}=2028.6016 \)
• \( (119 - 110.04)^{2}=(8.96)^{2}=80.2816 \)
• \( (145 - 110.04)^{2}=(34.96)^{2}=1222.2016 \)
• \( (114 - 110.04)^{2}=(3.96)^{2}=15.6816 \)
• \( (125 - 110.04)^{2}=(14.96)^{2}=223.8016 \)
• \( (114 - 110.04)^{2}=15.6816 \)
• \( (91 - 110.04)^{2}=(-19.04)^{2}=362.5216 \)
• \( (113 - 110.04)^{2}=(2.96)^{2}=8.7616 \)
• \( (125 - 110.04)^{2}=223.8016 \)
• \( (88 - 110.04)^{2}=(-22.04)^{2}=485.7616 \)
• \( (141 - 110.04)^{2}=(30.96)^{2}=958.5216 \)
• \( (105 - 110.04)^{2}=(-5.04)^{2}=25.4016 \)
• \( (116 - 110.04)^{2}=(5.96)^{2}=35.5216 \)
• \( (121 - 110.04)^{2}=(10.96)^{2}=120.1216 \)
• \( (186 - 110.04)^{2}=(75.96)^{2}=5769.9216 \)
• \( (136 - 110.04)^{2}=(25.96)^{2}=674.0016 \)
• \( (65 - 110.04)^{2}=2028.6016 \)
• \( (128 - 110.04)^{2}=(17.96)^{2}=322.5616 \)
• \( (107 - 110.04)^{2}=(-3.04)^{2}=9.2416 \)
• \( (97 - 110.04)^{2}=(-13.04)^{2}=169.9216 \)
• \( (126 - 110.04)^{2}=(15.96)^{2}=254.7216 \)
• \( (101 - 110.04)^{2}=(-9.04)^{2}=81.7216 \)
• \( (90 - 110.04)^{2}=(-20.04)^{2}=401.6016 \)
• \( (10 - 110.04)^{2}=(-100.04)^{2}=10008.0016 \)

Step3: Sum up all the \( (x_{i}-\bar{x})^{2} \) values

Let's add them one by one:
\( 167.9616+2028.6016 = 2196.5632 \)
\( 2196.5632+80.2816=2276.8448 \)
\( 2276.8448 + 1222.2016=3499.0464 \)
\( 3499.0464+15.6816 = 3514.728 \)
\( 3514.728+223.8016 = 3738.5296 \)
\( 3738.5296+15.6816 = 3754.2112 \)
\( 3754.2112+362.5216 = 4116.7328 \)
\( 4116.7328+8.7616 = 4125.4944 \)
\( 4125.4944+223.8016 = 4349.296 \)
\( 4349.296+485.7616 = 4835.0576 \)
\( 4835.0576+958.5216 = 5793.5792 \)
\( 5793.5792+25.4016 = 5818.9808 \)
\( 5818.9808+35.5216 = 5854.5024 \)
\( 5854.5024+120.1216 = 5974.624 \)
\( 5974.624+5769.9216 = 11744.5456 \)
\( 11744.5456+674.0016 = 12418.5472 \)
\( 12418.5472+2028.6016 = 14447.1488 \)
\( 14447.1488+322.5616 = 14769.7104 \)
\( 14769.7104+9.2416 = 14778.952 \)
\( 14778.952+169.9216 = 14948.8736 \)
\( 14948.8736+254.7216 = 15203.5952 \)
\( 15203.5952+81.7216 = 15285.3168 \)
\( 15285.3168+401.6016 = 15686.9184 \)
\( 15686.9184+10008.0016 = 25694.92 \)

Step4: Calculate the variance and then the standard deviation

The variance \( s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}=\frac{25694.92}{25 - 1}=\frac{25694.92}{24}\approx1070.6217 \)

The standard deviation \( s=\sqrt{1070.6217}\approx32.72 \)

Answer:

The standard deviation is \( \boldsymbol{32.72} \) (rounded to the nearest hundredth)