Question 8.4: Mobile Phone Service Provider Market Share Study A mobile ph......

1    The hypothesis test is classified as a two-tailed (or two-sided) test for a single proportion because:

• the random variable (i.e. a Cell D Mobile prepaid phone user or not) is categorical, with the sample proportion summarising the proportion of prepaid mobile phone users who subscribe to Cell D Mobile as their service provider
• the management question requires us to test if the true proportion (or percentage) of prepaid mobile phone users who use Cell D Mobile as a service provider is equal to a specified value only (i.e. 15%).

Step 1: Define the null and alternative hypotheses

H_0: π = 0.15 Market share = 15%, as claimed by Cell D Mobile.

H_1: π ≠ 0.15

The Cell D Mobile management claim of a 15% market share resides in the null hypothesis.

Step 2: Determine the region of acceptance of the null hypothesis

Given α = 0.01 (i.e. 1% level of significance), and since this is a two-tailed test, the region of acceptance will be defined by both a critical lower limit and a critical upper limit around H_0.

The z-distribution is always used for hypothesis tests for proportions. Thus the two critical z-limits are those z-values that identify a combined tail area of 1% in the z-distribution.

These critical z-limits are z-crit = ±2.58 (Table 1, Appendix 1). Thus the region of acceptance for H_0 is −2.58 ≤ z ≤ +2.58.

The decision rule for accepting or rejecting H_0 is then stated as follows.

• Accept H_0 if z-stat falls between −2.58 and +2.58 (inclusive).
• Reject H_0 if z-stat falls either below −2.58 or above +2.58.

These critical z-limits define how many standard errors (±2.58) the sample proportion can fall away from the null hypothesised population proportion before H_0 is rejected in favour of H_1.

Step 3: Calculate the sample test statistic (z–stat)

The sample evidence is represented by the sample proportion, p, which is defined as the proportion of prepaid mobile phone users who subscribe to Cell D Mobile. This statistic, p, must now be converted into the z-test statistic (z-stat) using Formula 8.3.

Given x = 42 (number of respondents who subscribe to Cell D Mobile) and n = 360:

and the standard error is:

Note: The null hypothesised population proportion (in this example, π = 0.15) is always used to compute the standard error for a single proportion (i.e. the denominator of z-stat).

Then the sample test statistic (z-stat) is calculated as follows:

z-stat = \frac{0.1167-0.15}{0.0188} =\frac{-0.0333}{0.0188} =-1.771

This z-stat value indicates that the sample proportion (p = 0.1167) lies 1.771 standard error units below the hypothesised population proportion of π = 0.15 (i.e. 15%).

Step 4: Compare the sample test statistic to the area of acceptance

This sample test statistic, z-stat, must now be compared to the decision rule (from Step 2) to decide if it is close enough to the null hypothesised population proportion to allow us to accept H_0.

The sample test statistic, z-stat = −1.771, lies inside the region of acceptance of −2.58 ≤ z ≤ +2.58. Refer to Figure 8.9, which shows the sample test statistic (z-stat) in relation to the regions of acceptance and rejection.

Step 5: Draw statistical and management conclusions

Statistical Conclusion

Since z-stat lies inside the region of acceptance, we accept (do not reject) H_0 at the 1% level of significance. The sample evidence is not convincing (or strong) enough to reject the null hypothesis at the 1% level of significance. The null hypothesis is therefore probably true.

Management Conclusion

It can be concluded, with 99% confidence, that Cell D Mobile’s claim about their market share of the prepaid mobile phone market being 15% is probably true. The competitor has no compelling sample evidence to refute Cell D Mobile’s claim that their market share is 15%.

2   Step 1: Define the null and alternative hypotheses

This hypothesis test can be classified as a one-sided lower-tailed test for a single proportion. The management question requires that a strict inequality relationship be tested (i.e. is π < 0.15?)

H_0: π ≥ 0.15

H_1: π ≥ 0.15     Cell D Mobile’s market share is strictly less than 15%.

The management question now resides in the alternative hypothesis.

Step 2: Determine the region of acceptance of the null hypothesis

Given α = 0.10 (i.e. 10% level of significance), and since this is a one-sided lower-tailed test, the region of acceptance will be defined by a critical lower limit only.

Using the z-distribution, the critical z-limit is that z-value that identifies a lower tail area of 10% in the z-distribution.

The critical z-limit is z-crit = −1.28 (Table 1, Appendix 1). Thus the region of acceptance for H_0 is z ≥ −1.28.

The decision rule for accepting or rejecting H_0 is then stated as follows:

• Accept H_0 if z-stat falls at or above –1.28.
• Reject H_0 if z-stat falls below –1.28.

Step 3: Calculate the sample test statistic (z-stat)

Since z-stat is based on sample evidence only and does not depend on whether the test is one-sided or two-sided, the z-stat value remains unchanged.

Thus z-stat = −1.771 (as for (a) above).

Step 4: Compare the sample test statistic to the area of acceptance

The sample test statistic, z-stat = −1.771 lies outside the region of acceptance of z ≥ −1.28. Refer to Figure 8.10, which shows the sample test statistic (z-stat) in relation to the regions of acceptance and rejection.

Step 5: Draw statistical and management conclusions

Statistical Conclusion

Since z-stat lies outside the region of acceptance, reject H_0 at the 10% level of significance.

There is strong sample evidence at the 10% level of significance to reject H_0 in favour of H_1. The alternative hypothesis is therefore probably true.

Management Conclusion

It can be concluded, with 90% confidence, that Cell D Mobile’s market share is significantly below 15%, and their claim cannot be supported based on the available market research evidence.