Inferences about Two Proportions technique (Chapter 9):
There is a claim that proportion of successes increased in 2009 compare to 2008.
For data given in Table 1
Table 1
Number of cases Number of successes
2008 400 180
2009 500 235
Calculate pooled sample proportion and z-test statistics
Use a significance level of á = 0.05 to make decision if there is a considerable change of proportion of successes.
This would fit a one-tailed proportional z-test (using proportions). The test would be one-tailed because of the alternate hypothesis (p2009 > p2008).
Here is one formula:
z = (p2009 - p2008)/√[pq(1/n1 + 1/n2)]
...where 'n' is the sample sizes, 'p' is (x1 + x2)/(n1 + n2), and 'q' is 1-p.
I'll get you started:
p = (180 + 235)/(400 + 500) = ? -->once you have the fraction, convert to a decimal (decimals are easier to use in the formula).
q = 1 - p
p2008 = 180/400
p2009 = 235/500
Convert all fractions to decimals. Plug those decimal values into the formula and find z. Once you have this value, compare to the critical value from a z-table using 0.05 significance for a one-tailed test.
If the test statistic exceeds the critical value from the table, reject the null and accept the alternate hypothesis (p2009 < p2008). If the test statistic does not exceed the critical value from the table, then you cannot reject the null and you cannot conclude a difference.
I hope this will help.
To calculate the pooled sample proportion and the z-test statistic, we need to follow these steps:
Step 1: Calculate the sample proportions for each year.
The sample proportion is calculated by dividing the number of successes by the number of cases.
For 2008:
Sample proportion (p1) = Number of successes / Number of cases = 180 / 400 = 0.45
For 2009:
Sample proportion (p2) = Number of successes / Number of cases = 235 / 500 = 0.47
Step 2: Calculate the overall pooled sample proportion.
The pooled sample proportion (p) is calculated by taking the weighted average of the sample proportions from both years.
p = [(Number of successes in 2008) + (Number of successes in 2009)] / [(Number of cases in 2008) + (Number of cases in 2009)]
p = (180 + 235) / (400 + 500) = 415 / 900 ≈ 0.461
Step 3: Calculate the standard error.
The standard error (SE) is calculated using the formula:
SE = √[(p * (1 - p) / n1) + (p * (1 - p) / n2)]
Where n1 and n2 are the number of cases in each year.
SE = √[(0.461 * (1 - 0.461) / 400) + (0.461 * (1 - 0.461) / 500)]
= √[(0.461 * 0.539 / 400) + (0.461 * 0.539 / 500)]
= √[(0.248979 / 400) + (0.248979 / 500)]
= √[0.000622 + 0.000498]
= √0.00112
≈ 0.0335
Step 4: Calculate the z-test statistic.
The z-test statistic is calculated using the formula:
z = (p1 - p2) / SE
z = (0.45 - 0.47) / 0.0335
= -0.02 / 0.0335
≈ -0.597
Step 5: Make a decision using the z-test statistic.
To make a decision, we compare the absolute value of the z-test statistic to the critical value for a significance level of α = 0.05.
The critical value for a two-tailed test at α = 0.05 is approximately 1.96.
Since the absolute value of -0.597 (|-0.597|) is less than 1.96, we do not reject the null hypothesis.
In conclusion, based on the given data and a significance level of α = 0.05, there is not enough evidence to conclude that there was a considerable change in the proportion of successes between 2008 and 2009.