Question

Tipping points in daily deal transactions? Online “daily deal” sites (e.g., Groupon) offer customers a voucher to purchase a product at discount prices. However, the number of voucher purchases must exceed a predetermined number before the deal becomes active. This key number is termed the “tipping point” in marketing. Characteristics of the tipping point were investigated in the Journal of Interactive Marketing (February 2016). A sample of 2,617 vouchers purchased from daily-deal sites in Korea had a mean tipping point of 112 sales with a standard deviation of 560 sales. The researchers want to estimate the true mean tipping point of all daily deal offerings in Korea with 95% confidence. Find and practically interpret this interval estimate.

Short answer

It is 95% confident that the true mean tipping point of all daily deal offerings in Korea lies between the interval 90.54 and 133.46.

Step-by-step solution

Given information

Let X is the tipping points.$\mathrm{n}=2617,\mathrm{\sigma }=560$and $\overline{\mathrm{x}}=112$.

Computing Confidence Interval

For the confidence level 95%, the level of significance is 0.95.

$\begin{array}{c}\left(1-\alpha \right)=0.95\\ \alpha =0.05\\ \frac{\alpha }{2}=0.025\end{array}$

From table, the value of$z_{\frac{\alpha }{2}}$ is given below:

$\begin{array}{c}{z}_{\frac{\alpha }{2}}=z_{0.025}\\ =1.96\end{array}$

Let, the confidence interval as,

$\begin{array}{c}\overline{x}\pm z_{\frac{\alpha }{2}}\left(\frac{\sigma }{\sqrt{n}}\right)=112\pm 1.96\left(\frac{560}{\sqrt{2617}}\right)\\ =112\pm 1.96\left(10.95\right)\\ =112\pm 21.46\end{array}$

That is $\left(112-21.46,112+21.46\right)=\left(90.54,133.46\right)$.

Therefore, the 90% confidence interval for$\mu$ is (90.54, 133.46).

Hence, it is 95% confident that the true mean tipping point of all daily deal offerings in Korea lies between the interval 90.54 and 133.46.