Question

For a particular radioactive sample, the total counting rate (sample plus background) was 450 cpm, and this value was obtained over a 15.0-min counting period. The background was counted for 2.0 min and gave 7 cpm. Estimate

(a) The corrected counting rate $R_c$

(b) The standard deviation associated with the corrected counting rate ${\sigma }_{R_c}$.

(c) The 95% confidence interval associated with the corrected counting rate.

Short answer

(a) The revised counting rate has a value of 443 cpm

(b) For the adjusted counting rate, the standard deviation ${\sigma }_R$is 5.79 cpm

(c) $R=443\pm 11cpm$is the value of the 95% confidence interval for the corrected counting rate.

Step-by-step solution

Part (a) Step 1. Given information

The revised counting rate will be calculated using the formula below:

$R_c=R_x-R_b\dots \dots \dots \left(1\right)$

Where,

$R_c=$correcting counting rate

$R_x=$counting rate for sample

$R_b=$counting rate for background

Total counting rate =$450\text{cpm}$

 Part (a) Step 2. Explanation 

$\begin{array}{r}{R}_x=450\mathrm{cpm}\\ R_b=7\mathrm{cpm}\end{array}$

therefore,

$\begin{array}{r}{R}_c=450\mathrm{cpm}-7\mathrm{cpm}\\ R_c=443\mathrm{cpm}\end{array}$

Part (b) Step 1 . Given information 

The formula for calculating the corrected counting rate's standard deviation is-

${\sigma }_{R_c}=\sqrt{\frac{R_x}{t_x}+\frac{R_b}{t_b}}$

Where,

$R_x=$Sample counting rate.

$R_b=$Background counting rate.

$t_x=$Time required for counting rate for sample.

$t_b=$Time required for background counting rate.

$\sigma R_c=$The corrected counting rate's standard deviation.

Part (b) Step 2. Explanation 

$\begin{array}{r}{R}_x=450\mathrm{cpm}\\ R_b=7\mathrm{cpm}\\ t_x=15\min \\ t_b=2\min \end{array}$

therefore,

$\begin{array}{r}{\sigma }_{R_c}=\sqrt{\frac{450}{15}+\frac{7}{2}}\\ {\sigma }_{R_c}=5.79\mathrm{cpm}\end{array}$

Part (c) Step 1. Given information 

The formula that will be employed-

$R=R\pm z{\sigma }_R$

The standard deviation for the corrected counting rate is calculated using the following formula:

${\sigma }_{R_c}=\sqrt{\frac{R_x}{t_x}+\frac{R_b}{t_b}}$

Where,

$R_x=$Sample counting rate

$R_b=$Background counting rate

$t_x=$Time necessary for sample counting rate

role="math" localid="1646035459199" $t_b=$Time required for background counting rate

$\sigma R_c=$The corrected counting rate's standard deviation

$R_c=R_x-R_b$

Part (c) Step2 . Explanation  

The formula for calculating the counting rate's confidence interval is as follows:

$R=R\pm z{\sigma }_R$

Given that-

$95\%$confidence interval for the revised counting rate

The $z$value for the $90\%$confidence interval is $=\pm 1.96$

Calculated value-
$\begin{array}{r}{R}_c=443\mathrm{cpm}\\ {\sigma }_{R_c}=5.79\mathrm{cpm}\end{array}$

Therefore,

$\begin{array}{r}R=443\pm 1.96\times 5.79\\ R=443\pm 11\mathrm{cpm}\end{array}$