A random sampleof size 40 is taken from a normally distributed population which has a mean mu and variance 3.5^2 . The sample mean is 16.5 . Test at the 1% level of significance the proposition that the population mean mu is qual to 15, with the alternative that it is not.
To test the hypothesis that the population mean μ is equal to 15, we will perform a one-sample t-test. Here's how you can carry out the test:
Step 1: State the null and alternative hypotheses.
- Null hypothesis (H0): μ = 15 (population mean is equal to 15)
- Alternative hypothesis (Ha): μ ≠ 15 (population mean is not equal to 15)
Step 2: Determine the level of significance.
In this case, the level of significance is given as 1% or 0.01. This means we will reject the null hypothesis if the probability (p-value) of observing the sample mean in a sample with a true population mean of 15 is less than 0.01.
Step 3: Compute the test statistic.
The test statistic for a one-sample t-test is given by:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
In this case, sample mean (x̄) = 16.5, hypothesized mean (μ₀) = 15, sample standard deviation (s) is unknown, but we can use the sample variance (s^2 = 3.5^2 = 12.25) to estimate it. The sample size (n) is 40.
t = (16.5 - 15) / (sqrt(12.25) / sqrt(40))
Step 4: Determine the critical value(s).
Since the alternative hypothesis is two-sided (μ ≠ 15), we need to find the critical values for a two-tailed test. The critical value is obtained from the t-distribution table or a statistical calculator using the degrees of freedom (n - 1) and the desired level of significance (0.01).
Step 5: Calculate the p-value.
The p-value is the probability of obtaining a sample mean as extreme as (or more extreme than) the observed sample mean, assuming the null hypothesis is true. We can calculate the p-value by determining the area in both tails of the t-distribution that falls beyond the critical values.
Step 6: Make a decision.
If the p-value is less than the level of significance (0.01), we will reject the null hypothesis. Otherwise, if the p-value is greater than or equal to 0.01, we will fail to reject the null hypothesis.
To summarize, the steps involved in testing the hypothesis that the population mean μ is equal to 15 are:
1. State the null and alternative hypotheses.
2. Determine the level of significance.
3. Compute the test statistic.
4. Determine the critical value(s).
5. Calculate the p-value.
6. Make a decision based on the p-value.
You can now follow these steps and perform the calculations to get the final result of your hypothesis test.