how can I decide to reject or fail to reject the following null hypothesis? let a= 0.05,data; p=0.69, n=32, test statistic z= 0.09095
H(0);p=0.68 H(1)p>0.68
To decide whether to reject or fail to reject the null hypothesis, we would compare the test statistic (z) to the critical value corresponding to the significance level (α).
First, determine the critical value using the significance level (α) = 0.05. Since this is a one-tailed test (H1: p > 0.68), we need to find the z-value related to the 0.05 in the tail. You can look up this value in a standard normal distribution table or use a calculator with a built-in function. The z-value corresponding to α = 0.05 in a one-tailed test is 1.645.
Now compare the test statistic (z) to the critical value:
Test statistic, z = 0.09095
Critical value = 1.645
Since the test statistic (0.09095) is less than the critical value (1.645), we fail to reject the null hypothesis.
In conclusion, there is not enough evidence to support the alternative hypothesis (p > 0.68) at a significance level of 0.05.
To decide whether to reject or fail to reject the null hypothesis, follow these steps:
Step 1: State the null and alternative hypotheses.
- Null Hypothesis (H0): p = 0.68
- Alternative Hypothesis (H1): p > 0.68
Step 2: Determine the significance level (a).
In this case, a is given as 0.05.
Step 3: Calculate the critical value or test statistic.
As the population proportion (p) is known, you can calculate the test statistic (z) using the formula:
z = (p - P0) / sqrt(P0 * (1 - P0) / n)
where P0 is the value stated in the null hypothesis.
Given:
p = 0.69
n = 32
P0 = 0.68
Calculating z:
z = (0.69 - 0.68) / sqrt(0.68 * (1 - 0.68) / 32)
z ≈ 0.09095
Step 4: Determine the critical value or rejection region.
Since the alternative hypothesis is p > 0.68, the critical value will be a right-tailed test.
Looking up the critical value for a z-test with a significance level of 0.05, we find that the critical value is approximately 1.645.
Step 5: Compare the test statistic with the critical value.
- If the test statistic is greater than the critical value, reject the null hypothesis.
- If the test statistic is less than or equal to the critical value, fail to reject the null hypothesis.
In this case, z = 0.09095, which is less than 1.645. Therefore, you fail to reject the null hypothesis.
Conclusion: Based on the given data and test statistic, there is not enough evidence to reject the null hypothesis that p = 0.68.
To decide whether to reject or fail to reject the null hypothesis, you need to perform a hypothesis test using the given information.
Here's how you can go about it:
Step 1: State the hypotheses
The null hypothesis (H0) is that the true population proportion (p) is equal to 0.68.
The alternative hypothesis (H1) is that the true population proportion (p) is greater than 0.68.
Step 2: Choose the significance level (α)
The significance level, denoted as α, determines how much evidence we need to reject the null hypothesis. In this case, it is given that α = 0.05, which is a commonly used level of significance.
Step 3: Determine the test statistic
The test statistic for testing a population proportion is the z-statistic, which is calculated using the formula:
z = (p - P) / √[(P(1 - P)) / n]
where p is the sample proportion, P is the hypothesized proportion under the null hypothesis, and n is the sample size.
In this case, the sample proportion (p) is given as 0.69, the hypothesized proportion (P) is 0.68, and the sample size (n) is 32. Plugging in these values, we get:
z = (0.69 - 0.68) / √[(0.68(1 - 0.68)) / 32]
z = 0.01 / 0.064
Therefore, the calculated test statistic (z) is approximately 0.15625.
Step 4: Determine the critical region
The critical region is the range of values that leads to rejecting the null hypothesis. For a one-tailed hypothesis test with significance level α = 0.05, we need to find the value of z that corresponds to the cumulative probability of 1 - α, or 0.95.
Using a standard normal distribution table or a statistical software, you can find that the critical z-value for a one-tailed test with 0.95 cumulative probability is approximately 1.645.
Step 5: Make a decision
If the calculated test statistic (z) falls in the critical region (i.e., z > 1.645), then we reject the null hypothesis. If the calculated test statistic falls outside the critical region (i.e., z ≤ 1.645), then we fail to reject the null hypothesis.
In this case, the calculated test statistic (z = 0.15625) is less than the critical value of 1.645. Therefore, we fail to reject the null hypothesis.
In conclusion, based on the given information and the calculated test statistic, we fail to reject the null hypothesis H0: p = 0.68.