魅影直播

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.

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.

That is, the proportion of people who receive a $1 bill and spend it is less than the proportion of people who receive 4quarters and spend the money.

The following formula is used to construct the confidence interval for estimating the difference in two population proportions:

\(CI = \left( {\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - E,\left( {{{\hat p}_1} - {{\hat p}_2}} \right) + E} \right)\)

Where,

• \({\hat p_1}\)denotes the sampleproportion of people who received a $1 bill and spent it.
• \({\hat p_2}\)denotes the sampleproportion of people who received 4 quarters and spent the money.
• E is the margin of error.

The margin of error has the following formula:

\(E = {z_{\frac{\alpha }{2}}}\sqrt {\frac{{{{\hat p}_1}{{\hat q}_1}}}{{{n_1}}} + \frac{{{{\hat p}_2}{{\hat q}_2}}}{{{n_2}}}} \)

The following values are obtained from the previous exercise:

\({\hat p_1} = 0.261\)

\({\hat p_2} = 0.628\)

\(\begin{array}{l}{n_1} = 46\\{n_2} = 43\end{array}\)

The level of significance is equal to 0.05. Thus, the corresponding confidence level is equal to 95%.

The value of\({z_{\frac{\alpha }{2}}}\)when\(\alpha = 0.05\)is equal to 1.96.

Thus, the margin of error is equal to:

\(\begin{aligned}{c}E &= {z_{\frac{\alpha }{2}}}\sqrt {\frac{{{{\hat p}_1}{{\hat q}_1}}}{{{n_1}}} + \frac{{{{\hat p}_2}{{\hat q}_2}}}{{{n_2}}}} \\ &= \left( {1.96} \right)\sqrt {\frac{{\left( {0.261} \right)\left( {1 - 0.261} \right)}}{{46}} + \frac{{\left( {0.628} \right)\left( {1 - 0.628} \right)}}{{43}}} \\ &= 0.192\end{aligned}\)

Thus, the 95% confidence interval is equal to:

\(\begin{aligned}{c}CI &= \left( {\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - E,\left( {{{\hat p}_1} - {{\hat p}_2}} \right) + E} \right)\\ &= \left( {\left( {0.261 - 0.628} \right) - 0.192,\left( {0.261 - 0.628} \right) + 0.192} \right)\\ &= \left( { - 0.559, - 0.175} \right)\end{aligned}\)

Therefore, the confidence interval to estimate the difference in the two proportions of people is equal to (鈥�0.559, 鈥�0.175).

If the constructed confidence interval contains the value 0, there is not enough evidence to support the given claim.

If the constructed confidence interval does not contain the value 0, there is enough evidence to support the given claim.

Thus, the property of whether the confidence interval contains 0 is used to reach the conclusions.

Here, the confidence interval of (鈥�0.559, 鈥�0.175) does not contain the value 0.

So, 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.