Question

Astronomers have recently discovered an unknown distant galaxy and noticed something abnormal. Specifically, they suspect the percentage of stars that host planets in this galaxy is lower than is typical. (It is known that around 75% of stars throughout the universe have planets). Given the current state of technology, astronomers are not able to conduct a complete scan of all the stars in this galaxy. So instead they randomly observed 45 stars within the new galaxy, and found only 24 of them had at least one planet. They would like for you to conduct a hypothesis test to verify their suspicion.

Write a sentence defining the relevant parameters from your selected pair of hypotheses.

Are the assumptions necessary for this hypothesis test satisfied? Justify your answer.

Calculate the appropriate test statistic for your hypothesis test, to three decimal places.

Suppose the value of the relevant test statistic were -4.45. (This is not the correct value of the test statistic.)

Clearly state your conclusion to the hypothesis (reject/fail to reject/accept

H0/HA) at an α=0.01 level, and write a sentence interpreting your conclusion in the context of the problem.

Answer 1

Here we have

n=45 and p=0.75

So np = 33.75 and n(1-p) = 11.25 both are greater than 10 so normal distribution can be assumed.

Conclusion: There is evidence to conclude at 1% level of significance that  the percentage of stars that host planets in this galaxy is lower than is typical.

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Since test statistics -4.45 is less than critical value -2.326 so we reject the null hypothesis.

Conclusion: There is evidence to conclude at 1% level of significance that  the percentage of stars that host planets in this galaxy is lower than is typical.