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, Which of the following is the correct null and alternative hypothesis? Select one: a. Ho: = 24 vs. HA: <24 o b. Ho : r = 24 vs. HA: << 24 c. Ho: = 0.75 vs. HA:< 0.75 O d. Ho: p=0.75 vs. HA: 0.75 e. Ho: p = 24/45 vs. HA:> 24/45 f. Ho: p = 0.75 vs. HA: p <0.75 o g. Ho:p<0.75 vs. HA: p=0.75 Write a sentence defining the relevant parameters from your selected pair of hypotheses. To type the mathematical symbols, you may refer to x as x_bar, as mu, and p asp_hat based on your selected hypotheses, if and as necessary. 7 i- B 1 E ls & !

Answer 1

The information is given as follows: The sample size that denotes the randomly observed stars within the new galaxy is n= 45 The number of stars that host at least one planet is x= 24. The sample proportion of stars that host at least one planet is calculated as follows: - - 24 45 = 0.533 Hence, the sample proportion of stars that host at least one planet is The hypothesis is given as follows: The null hypothesis is H.: p=0.75. The alternative hypothesis is HA: p<0.75 . Hence, the correct option is f). The assumptions need to be assumed that the population is normally distributed.

The test statistic is calculated as follows: 7- p-p p(1-p) 0.533-0.75 10.75(1-0.75) V 45 = -3.362 Hence, the test statistic is t=-3.362 The level of significance is a =0.01. The P-value of the test is calculated as follows: P-value = P(Z <-3.362) (From normal table P(Z <-3.362) = 0.000387) = 0.000387 Hence, P-value = 0.000387. Decision rule: If P-value<a , then reject the null hypothesis. Here, P-value = 0.000387 <a = 0.01, then reject the null hypothesis. Therefore, it can be concluded that the percentage of stars that host planets in this galaxy is lower than is typical.