Comparing purchasers and nonpurchasers of toothpaste.

Step 1 of 4

a) 

Let \({\mu _1}\) = Mean age of Purchasers and \({\mu _2}= Mean age of Non Purchasers.

The hypothesis of interest is,

\({H_0}:{\mu _1} = {\mu _2}\), i.e., there is no significant difference between the Mean age of Purchasers and Non-purchasers.

\({H_a}:{\mu _1} \ne {\mu _2}\), i.e., there was a significant difference between the Mean age of Purchasers and Non-purchasers.

Age of Purchasers (\({\overline x _1}\)):

The mean of sample 1 \({\overline x _1}\) is 39.8.

Age of Non purchasers (\({\overline x _2}\)):

The mean of sample 2 \({\overline x _2}\) is 47.2.

The pooled variance can be calculated as,

\(S_P^2 = \frac{{\sum {{{\left( {{x_1} - {{\overline x }_1}} \right)}^2}}  + \sum {{{\left( {{x_2} - {{\overline x }_2}} \right)}^2}} }}{{{n_1} + {n_2} - 2}}\) 

Here \({n_1},{n_2}\) are the sizes of the samples, which are 20. Substitute the values in the above formula, and we get,

\(S_P^2 = \frac{{1915.2 + 3525.2}}{{20 + 20 - 2}}\)

\(S_P^2 = \frac{{1915.2 + 3525.2}}{{38}}\)

\(S_P^2 = 143.17\)

The test statistic can be determined as,

\(t = \frac{{{{\overline x }_1} - {{\overline x }_2}}}{{\sqrt {S_P^2\left( {\frac{1}{{{n_1}}} + \frac{1}{{{n_2}}}} \right)} }}\) 

Substitute the values in the above formula, and we get,

\(t = \frac{{39.8 - 47.2}}{{\sqrt {143.17\left( {\frac{1}{{20}} + \frac{1}{{20}}} \right)} }}\)

\(t =  - 1.96\)

The degree of freedom is \({n_1} + {n_2} - 2 = 20 + 20 - 2 = 38\).

For df = 38 at\( \alpha  = 0.10\) and we find the critical value as \({t_{\alpha /2}} = 1.69\).

The value of the test statistic falls in the rejection region, and We reject the null hypothesis \({H_0}\). 

Hence, the data present has sufficient evidence to conclude there is a difference in the mean age of purchasers and non-purchasers.