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Задача 21. Среднее время сборки изделия составляло 90 минут. Инженер изобрел но...

Jun 21, 2024

Solution by Steps

step 1

First, we need to calculate the sample mean

\bar{x}

of the given data:

79, 74, 112, 95, 83, 96, 77, 84, 70, 90

step 2

The sample mean

\bar{x}

is calculated as follows:

\bar{x} = \frac{79 + 74 + 112 + 95 + 83 + 96 + 77 + 84 + 70 + 90}{10} = \frac{860}{10} = 86

minutes

step 3

Next, we calculate the sample standard deviation

s

. The formula for the sample standard deviation is

s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}

step 4

Calculate each

(x_i - \bar{x})^2

(79-86)^2 = 49

(74-86)^2 = 144

(112-86)^2 = 676

(95-86)^2 = 81

(83-86)^2 = 9

(96-86)^2 = 100

(77-86)^2 = 81

(84-86)^2 = 4

(70-86)^2 = 256

(90-86)^2 = 16

step 5

Sum these values:

49 + 144 + 676 + 81 + 9 + 100 + 81 + 4 + 256 + 16 = 1416

step 6

Calculate the sample variance:

\frac{1416}{10-1} = \frac{1416}{9} = 157.33

step 7

The sample standard deviation

s

\sqrt{157.33} \approx 12.54

minutes

step 8

To construct a 95% confidence interval for the mean, we use the formula

\bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}}

step 9

For a 95% confidence level and

n-1 = 9

degrees of freedom, the critical value

t_{\alpha/2}

from the t-distribution table is approximately 2.262

step 10

Calculate the margin of error:

E = t_{\alpha/2} \frac{s}{\sqrt{n}} = 2.262 \frac{12.54}{\sqrt{10}} \approx 8.98

step 11

The confidence interval is:

86 \pm 8.98

, which gives us

(77.02, 94.98)

Answer

The 95% confidence interval for the new mean assembly time is

(77.02, 94.98)

minutes.

Key Concept

Confidence Interval

Explanation

A confidence interval provides a range of values that is likely to contain the population mean with a certain level of confidence, in this case, 95%.