suppose you are testing Ho:p=.65 versus Ha: p<.65 . for a random sample of 100 people, x=58, where x denotes the number in the sample that have the characteristic of interest . use a .01 level of significance to test this hypothesis.
Null hypothesis:
Ho: p = .65 -->meaning: population proportion is equal to .65
Alternative hypothesis:
Ha: p < .65 -->meaning: population proportion is less than .65
Using a formula for a binomial proportion one-sample z-test with your data included, we have:
z = (.58 - .65) -->test value (58/100 = .58) minus population value (.65)
divided by
√[(.65)(.35)/100] --> .35 represents 1 - .65 and 100 is the sample size.
Use a z-table to find the critical or cutoff value for a one-tailed test (lower tail) at .01 level of significance. The test is one-tailed because the alternative hypothesis is showing a specific direction (less than).
If the test statistic exceeds the critical value you find from the table, reject the null. If the test statistic does not exceed the critical value from the table, do not reject the null.
You can draw your conclusions from there.
I hope this will help get you started.
To test the hypothesis H₀: p = 0.65 versus H₁: p < 0.65, where p is the population proportion, you can use the hypothesis testing procedure for the population proportion.
Step 1: Formulate the null and alternative hypotheses:
H₀: p = 0.65
H₁: p < 0.65
Step 2: Determine the test statistic.
Since the sample size is large (n = 100) and the conditions for using the normal distribution are satisfied (np ≥ 10 and n(1-p) ≥ 10), you can use the normal distribution to approximate the sampling distribution of the proportion.
The test statistic for testing a proportion is the z-score, which is calculated as follows:
z = (x - np₀) / √(np₀(1-p₀))
where x is the sample proportion, n is the sample size, np₀ is the expected number of successes under the null hypothesis, and p₀ is the hypothesized population proportion (0.65 in this case).
Step 3: Set the level of significance (α).
In this case, the significance level (α) is 0.01, which means we are willing to accept a 1% chance of making a Type I error (rejecting the null hypothesis when it is true).
Step 4: Calculate the test statistic and p-value.
Given that x = 58 and n = 100, we can calculate the test statistic as follows:
z = (58 - 100(0.65)) / √(100(0.65)(1-0.65))
Step 5: Determine the critical region or rejection region.
Since the alternative hypothesis is one-tailed (p < 0.65), we need to find the critical value from the standard normal distribution corresponding to an alpha level of 0.01.
Step 6: Make a decision and interpret the results.
If the test statistic falls within the critical region, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 7: Draw a conclusion.
Based on the decision made in Step 6, we draw a conclusion regarding the hypothesis and interpret the results in the context of the problem.
Note: I cannot provide the specific calculation for the test statistic and critical value without exact values, but you can use statistical software or a z-table to find these values.
To test the hypothesis Ho: p = 0.65 versus Ha: p < 0.65, where p represents the proportion of people who have the characteristic of interest, we can use a one-sample proportion test.
Step 1: Define the hypotheses:
Ho: p = 0.65 (Null hypothesis)
Ha: p < 0.65 (Alternative hypothesis)
Step 2: Determine the level of significance:
A .01 level of significance is given in the problem.
Step 3: Conduct the hypothesis test:
We can calculate the test statistic and compare it to the critical value to make a decision.
The test statistic for a one-sample proportion test is given by:
Z = (x - np) / sqrt(np(1-p))
Where:
x = number of people in the sample with the characteristic of interest
n = sample size
p = hypothesized proportion
Given:
x = 58
n = 100
p = 0.65
Calculating the test statistic:
Z = (58 - (100 * 0.65)) / sqrt(100 * 0.65 * (1 - 0.65))
Z = (58 - 65) / sqrt(100 * 0.65 * 0.35)
Z = -7 / sqrt(22.75)
Z ≈ -1.41
Step 4: Determine the critical value:
Since the alternative hypothesis is one-tailed (p < 0.65), we are interested in the left tail of the normal distribution.
Since we have a level of significance of .01, we need to find the critical Z-value that corresponds to that level of significance. Looking in the Z-table, we find that a Z-value of -2.33 corresponds to a cumulative probability of 0.01.
Step 5: Make a decision:
Since the test statistic (-1.41) is not less than the critical value (-2.33), we fail to reject the null hypothesis.
Therefore, based on the given data, there is not enough evidence to support the claim that the true proportion is less than 0.65, at a .01 level of significance.
Note: The specific critical value and test statistic values may vary slightly depending on the level of significance and rounding used.