I really need help with my statistical analysis... i cant understand it. Here are the questions
A sample group was surveyed to determine which of two brands of soap was preferred. H0 :p = 0.50; H1: p is not equal to 0.50. Thirty-eight of 60 people indicated a preference. At the .05 level of significance, we can conclude that:
The performance of students on a test resulted in a mean score of 25. A new test is instituted and the instructor believes the mean score is now lower. A random sample of 64 students resulted in 40 scores below 25. At a significance level of α = .05:
You can try a proportional one-sample z-test for the first problem since this problem is using proportions.
Here's a few hints to get you started:
Using a formula for a proportional one-sample z-test with your data included, we have:
z = .63 - .50 -->test value (38/60 is approximately .63) minus population value (.50)
divided by
√[(.50)(.50)/60]
Finish the calculation. Remember if the null is not rejected, then there is no difference. If the null is rejected (this is a two-tailed test), then there is a difference and p is not equal to .50.
For the second problem, do the same kind of test. The test proportion will be 40/64 or .625. The sample size is 64. Everything else will be the same as the first problem when plugging the data into the formula. Once you have the test statistic, you can draw your conclusions from there.
I've been trying to work out these problems for about an hour.. I was given multiple choice answers but cannot get my answers even near the ones that were given to me.
Here they are
First problem:
A. z = 0.75, fail to reject H0.
B. z = 1.94, fail to reject H0.
C. z = 1.94, reject H0.
D. z = 2.19, reject H0
second:
A. z = 3.75, we can reject the null hypothesis.
B. z = 1.875, we fail to reject the null hypothesis.
C. z = -1.625, we fail to reject the null hypothesis.
D. z = -1.875, we can reject the null hypothesis.
To solve these questions, we need to perform hypothesis tests. Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population using sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), collecting sample data, and performing statistical tests to determine if there is enough evidence to support the alternative hypothesis.
1. For the first question:
Null hypothesis (H0): p = 0.50 (The proportion of people who prefer the two brands of soap is equal)
Alternative hypothesis (H1): p is not equal to 0.50 (The proportion of people who prefer the two brands of soap is different)
To determine if we can conclude that the proportion of people who prefer the soap brands is different from 0.50, we calculate the test statistic and compare it to the critical value at a significance level of 0.05.
First, compute the test statistic:
- Convert the number of people who prefer the soap to a proportion: 38/60 = 0.6333
- Compute the test statistic: (observed proportion - expected proportion) / standard error of the proportion
- Test statistic = (0.6333 - 0.50) / sqrt((0.50 * (1 - 0.50)) / 60)
- Test statistic ≈ 2.506
Next, find the critical value at the significance level of 0.05. We compare the test statistic to the critical value based on the degrees of freedom (df) for this test. Since this is a two-tailed test, we divide the significance level (α) by 2 (0.025) to find each critical value.
Finally, compare the test statistic with the critical value:
- If the test statistic falls outside the critical value region, we reject the null hypothesis in favor of the alternative hypothesis. This means there is enough evidence to conclude that the proportion of people who prefer the soap brands is different from 0.50.
- If the test statistic falls within the critical value region, we fail to reject the null hypothesis. This means there is not enough evidence to conclude that the proportion of people who prefer the soap brands is different from 0.50.
2. For the second question:
Null hypothesis (H0): μ = 25 (The mean test score is equal to 25)
Alternative hypothesis (H1): μ < 25 (The mean test score is lower than 25)
To determine if we can conclude that the mean test score is lower than 25, we calculate the test statistic and compare it to the critical value at a significance level of 0.05.
First, compute the test statistic:
- Compute the sample mean: sum of sample scores / sample size
- Sample mean = 40 / 64 ≈ 0.625
- Compute the test statistic: (sample mean - population mean) / standard error of the mean
- Test statistic = (0.625 - 25) / (standard deviation of the population / sqrt(sample size))
- Test statistic ≈ -14.375
Next, find the critical value at the significance level of 0.05. Since this is a one-tailed test in the left tail, we can directly find the critical value associated with the 0.05 significance level.
Finally, compare the test statistic with the critical value:
- If the test statistic falls outside the critical value region, we reject the null hypothesis in favor of the alternative hypothesis. This means there is enough evidence to conclude that the mean test score is lower than 25.
- If the test statistic falls within the critical value region, we fail to reject the null hypothesis. This means there is not enough evidence to conclude that the mean test score is lower than 25.
Remember, hypothesis tests provide a way to make statistical conclusions based on sample data, but it is important to consider other factors and the context of the problem as well.