Question

Redeeming tickets from textbook dispatches. Numerous companies now use textbook messaging on cell phones to sell their products. One way to do this is to shoot repairable reduction pasteboard (called an m- pasteboard) via a textbook. The redemption rate of m- tickets — the proportion of tickets redeemed — was the subject of a composition in the Journal of Marketing Research (October 2015). In a two-time study, over boardwalk shoppers shared by subscribing up to admit m-voucher. The experimenters were interested in comparing the redemption rates of m- tickets for different products in a sample of m- tickets for products vended at a milk-shake. Store, 79 were redeemed; in a sample of m- tickets for products vended at a donut store, 72 were redeemed.

a. Cipher the redemption rate for the sample of milk-shake m- tickets.

b. Cipher the redemption rate for the sample of donut m- tickets.

c. Give a point estimate for the difference between the actual redemption rates.

d. Form a 90 confidence interval for the difference between the actual redemption rates. Give a practical interpretation of the output.

e. Explain the meaning of the expression “90 confident” in your answer to part d.

f. Grounded on the interval, part d, is there a “ statistically.” A significant difference between the redemption rates? (Recall that a result is “ statistically” significant if there is substantiation to show that the true difference in proportions isn't 0.)

g. Assume the true difference between redemption rates must exceed.01 ( i.e., 1) for the experimenters to consider the difference “ virtually” significant. Based on the interval, part c, is there a “virtually” significant difference between the redemption rates?

Short answer

The confidence interval (CI) is a set of values that are likely to encompass a population number with a given level of certainty.

Step-by-step solution

Step-by-Step Solution Step 1: (a) Find the redemption rate for the sample of the milk-shake m-voucher

Consider X_M = 79 and n_M=2,447.

The redemption proportion for the milk-shake m-voucher test is,

$\overline{P{¯}_M}=\frac{x_M}{n_M}$

$=\frac{79}{2447}$

$=0.032$

Thus, the redemption rate for the sample of the milk-shake m-voucher is 0.032.

(b) Find the redemption rate for the sample donut m-voucher

Consider x_D = 72 and n_D = 6,619.

The redemption rate for the sample of donut m-voucher is,

$$\begin{gathered} {\overline{P}}_M=\frac{x_D}{n_D} \\ =\frac{72}{6619} \end{gathered}$$

= 0.011

So, the reclamation rate for the sample of the donut m-voucher is 0.011.

(c) Find the point estimate for the difference between the actual redemption rates

$\overline{P}{}_M-\overline{P}{}_D=0.032-0.011$

= 0.021

Thus, the point estimate for the difference between the actual redemption rates is 0.021.

(d) Find the 90% confidence interval for the difference between the actual redemption rates

The critical value for a two-tailed test is obtained below:

Here, the test is two-tailed, and the significance level is $\alpha =0.10$

The rejection region for the two-tailed test is$\left|z_c\right|>z_{\frac{a}{2}}$.

The confidence coefficient is 0.90.

So,

(1-α) = 0.90

$\alpha$= 0.10

$\frac{\alpha }{2}$= 0.05

From Appendix D Table II, the critical value for the two-tailed test with α = 0.10 is α =0.10 is $z_{\frac{a}{2}\left(-0.05\right)}$$z_{\frac{a}{2}\left(-0.05\right)}$= ±1.645. so, the refusal region is $\left|z_c\right|$ > 1.645.

(e) 90% confidence interval

The 90% confidence interval is obtained below:

$\left[\frac{({\overline{P}}_M-{\overline{P}}_D)\pm z_{0.05}}{\sqrt{\frac{{\overline{P}}_M(1-{\overline{P}}_M)}{n_M}+\frac{{\overline{P}}_D(1-{\overline{P}}_0)}{n_D}}}\right]=0.021\pm 1.645\sqrt{\frac{0.032(1-0.032)}{2447}+\frac{0.011(1-0.011)}{6619}}$

$$\begin{gathered} =0.021\pm 1.645(0.00378) \\ =0.021\pm 0.006 \\ =(0.015,0.027) \end{gathered}$$

Thus, the 90% confidence interval is (0.015, 0.027).

Interpretation:

There is 90% confidence that the difference between the actual redemption rates between milk-shake m-voucher and donut m-voucher lies between 0.015 and 0.027.

Explanation:

The 90% of all similarly generated confidence intervals will contain the true value of the difference in redemption rates in repeated sampling.

(f) check the statistically significant difference between the reclamation rates or not

Rule:

If the hypothesized value lies outside the corresponding 100(1-α) % confidence interval, then reject the null hypothesis.

Here, the 90% confidence interval is (0.015. 0.827), which does not contain the hypothesized value of 0. The hypothesized value 0 lies outside the interval (0.015, 0.027). Thus, it can be concluded that the null hypothesis H₀ is rejected at = 0.10.

Hence, there is a statistically significant difference between the redemption rates.

(g) check the practically significant difference between the reclamation rates or not

If the genuine disparity among redeeming rates is more than 0.01, the disparity is considered "practically important."

Thus, there’s a "practically significant difference between the reclamation rates because the hypothesized value 0.01 lies outside the interval (0.015, 0.027).