Question

Denomination Effect In the article “The Denomination Effect” by Priya Raghubir and Joydeep Srivastava, Journal of Consumer Research, Vol. 36, researchers reported results from studies conducted to determine whether people have different spending characteristics when they have larger bills, such as a \(20 bill, instead of smaller bills, such as twenty \)1 bills. In one trial, 89 undergraduate business students from two different colleges were randomly assigned to two different groups. In the “dollar bill” group, 46 subjects were given dollar bills; the “quarter” group consisted of 43 subjects given quarters. All subjects from both groups were given a choice of keeping the money or buying gum or mints. The article includes the claim that “money in a large denomination is less likely to be spent relative to an equivalent amount in smaller denominations.” Test that claim using a 0.05 significance level with the following sample data from the study.

Short answer

There is enough evidence to support the claim that money in a large denomination is less likely to be spent relative to the same amount of money in smaller denominations.

Step-by-step solution

Given information

The number of people who were given a $1 bill/4 quarters and who spent the money is tabulated along with the individual sample sizes.

Hypotheses

It is claimed that money in a large denomination is less likely to be spent relative to the same amount of money in smaller denominations.

Since the given claim does not have an equality sign, the following hypotheses are set up:

Null Hypothesis: The proportion of people who spend the money is equal and does not depend on its denomination.

\({H_0}:{p_1} = {p_2}\)

Alternative Hypothesis: The proportion of people who spend the money when they receive a large denomination ($1 bill) is less than the proportion of people who spend the money when they receive smaller denominations (4 quarters).

\({H_1}:{p_1} < {p_2}\)

The test is left-tailed.

If the absolute value of the test statistic is greater than the critical value, the null hypothesis is rejected.

Important values

Let\({\hat p_1}\)denote the sampleproportion of people who received a $1 bill and spent it.

\(\begin{aligned}{c}{{\hat p}_1} &= \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{people}}}}{{{\rm{Sample}}\;{\rm{size}}}}\\ &= \frac{{{x_1}}}{{{n_1}}}\\ &= \frac{{12}}{{46}}\\ &= 0.261\end{aligned}\)

Let\({\hat p_2}\)denote the sampleproportion of people who received 4 quarters and spent the money.

\(\begin{aligned}{c}{{\hat p}_2} &= \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{people}}}}{{{\rm{Sample}}\;{\rm{size}}}}\\ &= \frac{{{x_2}}}{{{n_2}}}\\ &= \frac{{27}}{{43}}\\ &= 0.628\end{aligned}\)

\({n_1}\)is equal to 46 and\({n_2}\)is equal to 43.

The value of the pooled sample proportion is computed as follows:

\(\begin{aligned}{c}\bar p &= \frac{{{x_1} + {x_2}}}{{{n_1} + {n_2}}}\\ &= \frac{{12 + 27}}{{46 + 43}}\\ &= 0.438\end{aligned}\)

\(\begin{aligned}{c}\bar q &= 1 - \bar p\\ &= 1 - 0.438\\ &= 0.562\end{aligned}\)

Compute the test statistic, critical value and p-value

The value of the test statistic is computed below:

\(\begin{aligned} z &= \frac{{\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - \left( {{p_1} - {p_2}} \right)}}{{\sqrt {\frac{{\bar p\bar q}}{{{n_1}}} + \frac{{\bar p\bar q}}{{{n_2}}}} }}\;\;\;\;{\rm{where}}\left( {{p_1} - {p_2}} \right) &= 0\\ &= \frac{{\left( {0.261 - 0.628} \right) - 0}}{{\sqrt {\frac{{\left( {0.438} \right)\left( {0.562} \right)}}{{46}} + \frac{{\left( {0.438} \right)\left( {0.562} \right)}}{{43}}} }}\\ &= - 3.49\end{aligned}\)

Thus, z= –3.49.

Using the standard normal table, the critical value of z corresponding to\(\alpha = 0.05\)for a left-tailed test is equal to –1.645.

The corresponding p-value is equal to 0.0002 which is obtained from the standard normal table using the test statistic (–3.49).

Since the absolute z-score value is greater than the critical value and the p-value is less than 0.05, the null hypothesis is rejected.

Conclusion

There is enough evidence to support the claim thatmoney in a large denomination is less likely to be spent relative to the same amount of money in smaller denominations.