Question

Suppose you doubt the assumption that the mean age of the stars is 3.3 billion years, but you don't know whether the true mean age is less or greater than 3.3 billion years.

Required:
a. To test whether the population mean is 3.3 billion years, what would your null and alternative hypotheses be?
b. What test will you use, how will the test work, and what are the conditions necessary to use the test? Does your situation meet those conditions?
c. Calculate yoour test statistic and P-value. Show your work, including the formulas you used to calculate the statistic.

Answer 1

(a) Null Hypothesis,
`H_0` :
`\mu` = 3.3 billion years

Alternate Hypothesis,
`H_A` :
`\mu\\eq` 3.3 billion years

(b) The conditions necessary to use this test is that the data must follow a normal distribution and we know about the population standard deviation.

(c) The value of z-test statistics is 1.77 and the P-value is 0.0768.

Explanation:

The complete question is: A theory predicts that the mean age of stars within a particular type of star cluster is 3.3 billion years, with a standard deviation of 0.4 billion years. (Their ages are approximately normally distributed.) You think the mean age is actually greater, and that this would lend support to an alternative theory about how the clusters were formed. You use a computer to randomly select the coordinates of 50 stars from the catalog of known stars of the type you're studying and you estimate their ages. You find that the mean age of stars in your sample is 3.4 billion years.

Suppose you doubt the assumption that the mean age of the stars is 3.3 billion years, but you don't know whether the true mean age is less or greater than 3.3 billion years.

Let
`\mu` = population mean age of the stars

(a) Null Hypothesis,
`H_0` :
`\mu` = 3.3 billion years {means that the population mean is 3.3 billion years}

Alternate Hypothesis,
`H_A` :
`\mu\\eq` 3.3 billion years {means that the population mean is different from 3.3 billion years, i.e. the true mean age is less or greater than 3.3 billion years}

(b) The test statistics that will be used here is One-sample z-test statistics because we know about the population standard deviation;

T.S. =
`(\bar X-\mu)/((\sigma)/(√(n) ) )` ~ N(0,1)

where,
`\bar X` = sample mean age of stars = 3.4 billion years

`\sigma` = population standard deviation = 0.4 billion years

n = sample of stars = 50

The conditions necessary to use this test is that the data must follow a normal distribution and we know about the population standard deviation. And the conditions are satisfied here.

(c) So, the test statistics =
`(3.4-3.3)/((0.4)/(√(50) ) )`

= 1.77

The value of z-test statistics is 1.77.

Also, the P-value of the test statistics is given by;

P-value = P(Z > 1.77) = 1 - P(Z
`\leq` 1.77)

= 1 - 0.9616 = 0.0384

For the two-tailed test, the P-value is calculated as =
`2 * 0.0384` = 0.0768.