College

a) The mean age of graduate students at a university is at most 31 years with a standard deviation of two years. A random sample of 15 graduate students is taken. The sample mean is 32 years and the sample standard deviation is three years. Are the data significant at the 1% level? The p-value is 0.0264.

- State the null and alternative hypotheses.
- Interpret the p-value.
- Write information given with correct symbols.

b) A bottle of water is labeled as containing 16 fluid ounces of water. You believe it is less than that. What type of test would you use?

c) Assume the null hypothesis states that the mean is at most 12. Is this a left-tailed, right-tailed, or two-tailed test?

d) What should you do if \(\alpha = \text{p-value}\)?

Answer :

Answer:

A.)

H0: μ ≤ 31

H1: μ > 31

B.)

H0: μ ≥ 16

H1: μ < 16

C.)

Right tailed test

D.)

If Pvalue is less than or equal to α ; we reject the Null

Step-by-step explanation:

The significance level , α = 0.01

The Pvalue = 0.0264

The decision region :

Reject the null if :

Pvalue < α

0.0264 > 0.01

Since Pvalue is greater than α ; then, we fail to reject the Null ;

Then there is no significant evidence that the mean graduate age is more Than 31.

B.)

H0: μ ≥ 16

H1: μ < 16

Null Fluid contains 16

Alternative hypothesis, fluid contains less than 16

One sample t - test

C.)

Null hypothesis :

H0 : μ ≤ 12

. The direction of the sign in the alternative hypothesis signifies the type of test or tht opposite direction of the sign in the null hypothesis.

Hence, this is a right tailed test ; Alternative hypothesis, H1 : μ > 12

d.)

If Pvalue is less than or equal to α ; we reject the Null.

Final answer:

The mean age of graduate students is significantly greater than 31 years, according to the data. A one-sample t-test should be used to test if the mean volume of water in the bottle is less than the labeled amount. The null hypothesis states that the mean is at most 12, making it a left-tailed test. If α (significance level) is equal to the p-value, the decision on rejecting or not rejecting the null hypothesis should be based on the predetermined level of significance.

Explanation:

a) The null hypothesis is that the mean age of graduate students is equal to or less than 31 years. The alternative hypothesis is that the mean age is greater than 31 years. The p-value is 0.0264, which is less than the significance level of 0.01. Therefore, we can reject the null hypothesis and conclude that the data is significant at the 1% level, indicating that the mean age of the graduate students is likely greater than 31 years.

b) You would use a one-sample t-test to test if the mean volume of water in the bottle is less than the labeled amount. This is because you are comparing a sample mean to a known population value.

c) This is a left-tailed test because the null hypothesis states that the mean is at most 12, implying that we are interested in determining if the mean is significantly less than 12.

d) If α (significance level) is equal to the p-value, it means that the obtained test statistic is exactly on the borderline of statistical significance. In other words, it is exactly at the threshold of rejecting or not rejecting the null hypothesis. In this case, the decision to reject or not reject the null hypothesis should be based on the predetermined level of significance, rather than solely on the p-value.

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