A survey of 100 mayonnaise purchasers showed that 65 were loyal to one brand. For 100 bath soap purchasers, only 53 were loyal to one brand. a) Perform a two-tailed test comparing the proportion of brand-loyal customers at á=.05.b) Form a confidence interval for the difference of proportions, without pooling the samples. Does it include zero? Could you please show me how you get this answer..
To perform a two-tailed test comparing the proportion of brand-loyal customers at a significance level of α = 0.05, we need to determine if there is a significant difference between the two proportions. We'll follow these steps:
a) Perform a two-tailed test:
Step 1: State the null and alternative hypotheses.
Null hypothesis (H0): The proportions are equal.
Alternative hypothesis (H1): The proportions are not equal.
Step 2: Calculate the test statistic.
The test statistic for comparing proportions is the z-statistic, given by:
z = (p1 - p2) / √(p*(1-p)*(1/n1 + 1/n2))
Where:
- p1 and p2 are the sample proportions (65/100 and 53/100, respectively).
- p is the pooled proportion, but since we're not pooling the samples, we'll use the overall proportion: p = (x1 + x2) / (n1 + n2).
- n1 and n2 are the sample sizes (100 for both).
Applying the formula:
z = (65/100 - 53/100) / √((65+53)/(100+100) * (1 - (65+53)/(100+100)) * (1/100 + 1/100))
z = (0.65 - 0.53) / √(0.1189)
z ≈ 3.39
Step 3: Determine the critical value(s) or p-value.
Since this is a two-tailed test, we'll need to consider both tails.
With α = 0.05, each tail has an area of α/2 = 0.025. We can find the critical value using a z-table or calculator, which corresponds to a cumulative probability of 0.975 (1 - 0.025) for each tail.
The critical value (CV) for a two-tailed test with α = 0.05 is approximately ±1.96.
Step 4: Make a decision.
Since the test statistic (3.39) is greater than the critical value (±1.96), we reject the null hypothesis. There is evidence to suggest a significant difference between the proportions of brand-loyal customers for mayonnaise and bath soap.
b) For the confidence interval without pooling the samples:
A confidence interval is a range of values that estimates the true difference between proportions. Without pooling the samples, we can use the formula:
CI = (p1 - p2) ± z * √((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))
Where the variables have the same meanings as in the previous explanation.
Plugging in the values:
CI = (65/100 - 53/100) ± 1.96 * √((65/100 * (1 - 65/100) / 100) + (53/100 * (1 - 53/100) / 100))
CI ≈ 0.12 ± 1.96 * √(0.001105 + 0.00099059)
CI ≈ 0.12 ± 0.0478
The confidence interval is approximately (0.0722, 0.1678) or 7.22% to 16.78%.
To determine if the confidence interval includes zero:
Since zero (0) is not within the interval (0.0722, 0.1678), we can conclude that there is a significant difference between the proportions of brand-loyal customers for mayonnaise and bath soap.