Question

a. Let aand bbe constants and let\({y_i} = a{x_i} + b\)fori=1, 2, . . . , n. What are the relationships between\(\bar x\)and\(\bar y\)and between\(s_x^2\)and\(s_y^2\)?

b. A sample of temperatures for initiating a certain chemical reaction yielded a sample average (°C) of 87.3 and a sample standard deviation of 1.04. What are the sample average and standard deviation measured in °F? (Hint:\(F = \frac{9}{5}C + 32\))

Short answer

a.

The relationship between\(\bar x\)and\(\bar y\)is:\(\bar y = a\bar x + b\).

The relationship between\(s_x^2\)and\(s_y^2\)is:\(s_y^2 = {a^2}s_x^2\).

b.

The sample average is 189.14 Fahrenheit.

The sample standard deviation is 1.872 Fahrenheit.

Step-by-step solution

Given information

There are two constants a and b.

\({y_i} = a{x_i} + b\)for i=1,2,…,n.

The average temperature for initiating a certain chemical reaction is 87.3 degrees Celsius.

The sample standard deviation is 1.04.

Describing the relationship between the sample means

The sample mean of x is given as,

\(\bar x = \frac{{\sum {{x_i}} }}{n}\)

The sample of y is given as,

\(\begin{array}{c}\bar y &=& \frac{{{y_1} + {y_2} + ... + {y_n}}}{n}\\ &=& \frac{{\left( {a{x_1} + b} \right) + \left( {a{x_2} + b} \right) + ... + \left( {a{x_n} + b} \right)}}{n}\\ &=& \frac{{a\left( {{x_1} + {x_2} + ... + {x_n}} \right) + nb}}{n}\\ &=& a\frac{{\sum {{x_i}} }}{n} + b\\ &=& a\bar x + b\end{array}\)

Thus, the relationship between \(\bar x\) and \(\bar y\) is \(\bar y = a\bar x + b\).

Describing the relationship between the sample variance

The sample variance of x is given as,

\(s_x^2 = \frac{{\sum {{{\left( {{x_i} - \bar x} \right)}^2}} }}{{n - 1}}\)

The sample variance of y is given as,

\(\begin{array}{c}s_y^2 &=& \frac{{\sum {{{\left( {{y_i} - \bar y} \right)}^2}} }}{{n - 1}}\\ &=& \frac{{\sum {{{\left( {a{x_i} + b - \left( {a\bar x + b} \right)} \right)}^2}} }}{{n - 1}}\\ &=& \frac{{\sum {{{\left( {a{x_i} + b - \left( {a\bar x + b} \right)} \right)}^2}} }}{{n - 1}}\\ &=& \frac{{\sum {{{\left( {a{x_i} - a\bar x} \right)}^2}} }}{{n - 1}}\\ &=& \frac{{{a^2}\sum {{{\left( {{x_i} - \bar x} \right)}^2}} }}{{n - 1}}\\ &=& {a^2}s_x^2\end{array}\)

Therefore, the relationship between \(s_x^2\) and \(s_y^2\) is: \(s_y^2 = {a^2}s_x^2\).

Computing the average in Fahrenheit.

Fahrenheit in Celsius is given as,

\(F = \frac{9}{5}C + 32\)

Since the average is affected by the change of origin and scale.

The sample average measured in Fahrenheit is computed as,

\(\frac{9}{5}\left( {87.3} \right) + 32 = 189.14\)

Therefore, the sample average is 189.14 Fahrenheit.

Computing the standard deviation in Fahrenheit.

Since the standard deviation is affected by only the change of scale.

The sample standard deviation measured in Fahrenheit is computed as,

\(\frac{9}{5}\left( {1.04} \right) = 1.872\)

Therefore, the sample standard deviation is 1.872 Fahrenheit.