Question

Various temperature measurements are recorded at different times for a particular city. the mean of 20°c is obtained for 60 temperatures on 60 different days. assuming that the population standard deviation is 1.5 degrees c, test the claim that the population mean is 22 degrees c. use a 0.05 significance level.

Answer 1

To test the claim that the population mean is 22 degrees C, use a one-sample t-test. Calculate the t-value and compare it to the critical t-value. If the calculated t-value is less than the critical t-value, reject the null hypothesis.

Step-by-step explanation:

To test the claim that the population mean is 22 degrees Celsius, we can use a one-sample t-test. The null hypothesis (H0) is that the population mean is equal to 22 degrees Celsius, and the alternative hypothesis (Ha) is that the population mean is not equal to 22 degrees Celsius. We will use a significance level of 0.05.

To perform the t-test, we can calculate the t-value using the formula:

t = (sample mean - hypothesized population mean) / (sample standard deviation / sqrt(sample size))

Substituting the given values:

t = (20 - 22) / (1.5 / sqrt(60))

t = -2 / (1.5 / sqrt(60))

t = -2 / 0.1932

t ≈ -10.35

Comparing the calculated t-value with the critical t-value at a significance level of 0.05 and degrees of freedom (df = sample size - 1), we find that the critical t-value is approximately -2.001. Since the calculated t-value (-10.35) is less than the critical t-value (-2.001), we reject the null hypothesis and conclude that there is strong evidence to support the claim that the population mean is different from 22 degrees Celsius.

Answer 2

To test the claim that the population mean is 22 degrees C, we will conduct a hypothesis test using a z-test. The calculated test statistic does not fall within the critical region, so we fail to reject the null hypothesis. Therefore, there is not enough evidence to support the claim that the population mean is different from 20°C at the 5% significance level.

Step-by-step explanation:

To test the claim that the population mean is 22 degrees C, we will conduct a hypothesis test.

Step 1: State the hypotheses.

• Null hypothesis (H0): The population mean is 20°C.
• Alternative hypothesis (Ha): The population mean is different from 20°C.

Step 2: Set the significance level.

The significance level is given as 0.05 (5%).

Step 3: Calculate the test statistic.

Since we are assuming that the population standard deviation is known (1.5 degrees C) and the sample size is large (60), we can use a z-test. The test statistic (z) can be calculated as follows:

z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))

Substituting the given values, we have:

z = (20 - 22) / (1.5 / sqrt(60))

Step 4: Determine the critical value.

Since the alternative hypothesis is two-sided, we need to find the critical values for a two-tailed z-test at the 5% significance level. The critical values for a two-tailed z-test at alpha/2 = 0.025 are approximately -1.96 and 1.96.

Step 5: Make a decision.

If the test statistic falls within the critical region (i.e., z < -1.96 or z > 1.96), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Step 6: State the conclusion.

In this case, based on the calculated test statistic, z, we do not reject the null hypothesis. There is not enough evidence to support the claim that the population mean is different from 20°C at the 5% significance level.