Questions: Churning, the loss of customers to a competitor, is especially a problem for telecommunications companies. Market researchers at a telecommunications company collect data from a sample of 5,526 subscribers. Of 2,750 basic subscribers selected, 868 churned in the last month. Of 2,776 premium subscribers selected, 856 churned in the last month. Complete parts (a) through (c). a. At the 0.05 level of significance, is there evidence of a difference in the proportion of basic and premium subscribers who churned? Let group 1 be the basic subscribers, and let group 2 be the premium subscribers. State the null and alternative hypotheses. Choose the correct answer. A. H0: π1 = π2 B. H0: π1 ≠ π2 H1: π1 ≠ π2 C. H0: π1 ≥ π2 D. H0: π1 < π2 H1: π1 < π2 H1: π1 ≥ π2 E. H0: π1 > π2 F. H0: π1 ≤ π2 H1: π1 ≤ π2 H1: π1 > π2

Churning, the loss of customers to a competitor, is especially a problem for telecommunications companies. Market researchers at a telecommunications company collect data from a sample of 5,526 subscribers. Of 2,750 basic subscribers selected, 868 churned in the last month. Of 2,776 premium subscribers selected, 856 churned in the last month. Complete parts (a) through (c).

a. At the 0.05 level of significance, is there evidence of a difference in the proportion of basic and premium subscribers who churned?

Let group 1 be the basic subscribers, and let group 2 be the premium subscribers. State the null and alternative hypotheses. Choose the correct answer.

A. H0: π1 = π2
B. H0: π1 ≠ π2 H1: π1 ≠ π2
C. H0: π1 ≥ π2
D. H0: π1 < π2 H1: π1 < π2 H1: π1 ≥ π2
E. H0: π1 > π2
F. H0: π1 ≤ π2 H1: π1 ≤ π2 H1: π1 > π2
Transcript text: Churning, the loss of customers to a compettor, is especially a problem for telecommunications companies Market researchers at a telecommunications company collect data from a sample of 5,526 subscribers. Of 2,750 basic subscribers selected, 868 churned in the last month Of 2,776 premum subscribers selected, 856 churned in the last month Complele parts (a) through (c) a. At the 0.05 level of significance, is there evidence of a difference in the proportion of basic and premium subscribers who churned? Let group 1 be the basic subscribers, and let group 2 be the premium subscribers. State the null and alternative hypotheses Choose the correct answer. A. $H_{0} \pi_{1}=\pi_{2}$ B. $H_{0} \pi_{1} \neq \pi_{2}$ $H_{1}: \pi_{1} \neq \pi_{2}$ $H_{1} \pi_{1}=\pi_{2}$ C. $H_{0} \pi_{1} \geq \pi_{2}$ D. $H_{0} \pi_{1}<\pi_{2}$ $H_{1} \pi_{1}<\pi_{2}$ $H_{1} \pi_{1} \geq \pi_{2}$ E. $H_{0} \cdot \pi_{1}>\pi_{2}$ F. $H_{0} \pi_{1} \leq \pi_{2}$ $\mathrm{H}_{1} \pi_{1} \leq \pi_{2}$ $H_{1} \pi_{1}>\pi_{2}$
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Solution

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Solution Steps

Step 1: Calculate Sample Proportions

For the basic subscribers, the sample proportion of churned subscribers is calculated as follows:

\[ \hat{p}_1 = \frac{868}{2750} \approx 0.3156 \]

For the premium subscribers, the sample proportion is:

\[ \hat{p}_2 = \frac{856}{2776} \approx 0.3084 \]

Step 2: Compute the Confidence Interval for the Difference in Proportions

We will calculate the confidence interval for the difference in proportions \( \hat{p}_1 - \hat{p}_2 \) at a 95% confidence level. The formula for the confidence interval is given by:

\[ (\hat{p}_1 - \hat{p}_2) \pm z \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}} \]

Where \( z \) is the critical value for a 95% confidence level, which is approximately \( 1.96 \).

Substituting the values:

\[ \hat{p}_1 - \hat{p}_2 = 0.3156 - 0.3084 = 0.0072 \]

Now, we calculate the standard error:

\[ \sqrt{\frac{0.3156(1 - 0.3156)}{2750} + \frac{0.3084(1 - 0.3084)}{2776}} \approx 0.0124 \]

Thus, the confidence interval becomes:

\[ 0.0072 \pm 1.96 \cdot 0.0124 \]

Calculating the margin of error:

\[ 1.96 \cdot 0.0124 \approx 0.0243 \]

Therefore, the confidence interval is:

\[ (0.0072 - 0.0243, 0.0072 + 0.0243) = (-0.0172, 0.0317) \]

Step 3: Interpret the Confidence Interval

The confidence interval for the difference in proportions is:

\[ (-0.0172, 0.0317) \]

Since this interval includes zero, we conclude that there is no significant difference in the proportion of churned subscribers between basic and premium plans.

Final Answer

There is no significant difference in the proportion of churned subscribers between basic and premium plans. The answer is:

\(\boxed{\text{There is no significant difference.}}\)

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