Why do we use \( \hat{p} \) as our estimates of both \( p_{1} \) and \( p_{2} \) ? Because it is the average of \( \hat{p}_{1} \) and \( \hat{p}_{2} \). Because it is easier than using \( \hat{p}_{1} \) and \( \hat{p}_{2} \) to estimate \( p_{1} \) and \( p_{2} \), respectively. Because we assume \( H_{0}: p_{1}=p_{2} \) is true.

Suppose the mean IQ score of people in a certain country is 100 . Suppose the director of a college obtains a simple random sample of 42 students from that country and finds the mean IQ is 103.6 with a standard deviation of 13.5 . Complete parts (a) through (d) below. (a) Consider the hypotheses \( \mathrm{H}_{0}: \mu=100 \) versus \( \mathrm{H}_{1}: \mu>100 \). Explain what the director is testing. Perform the test at the \( \alpha=0.01 \) level of significance. Write a conclusion for the test. Explain what the director is testing. Choose the correct answer below. A. The director is testing if the sample provided sufficient evidence that the population mean IQ score is actually greater than 100 . C. The director is testing if the sample provided sufficient evidence that the population mean IQ score is actually equal to 100 . D. The director is testing if the sample provided sufficient evidence that the population mean IQ score is actually not equal to 100 . Find the test statistic for this hypothesis test. \( \square \) (Round to two decimal places as needed.)

If the number 31 is added to the data set \( 3,5,6,6,8,11,12,22,26 \), by what value will the average of the data set increase?

(b) Consider the hypotheses \( \mathrm{H}_{0}: \mu=101 \) versus \( \mathrm{H}_{1}: \mu>101 \). Explain what the director is testing. Perform the test at the \( \alpha=0.01 \) level of significance. Write a conclusion for the test Explain what the director is testing. Choose the correct answer below. A. The director is testing if the sample provided sufficient evidence that the population mean IQ score is actually not equal to 101 . C. The director is testing if the sample provided sufficient evidence that the population mean IQ score is actually equal to 101 . D. The director is testing if the sample provided sufficient evidence that the population mean IQ score is actually not greater than 101 . Find the test statistic for this hypothesis test. 1.26 (Round to two decimal places as needed.) Find the P -value for this hypothesis test. (Round to three decimal places as needed.)

5 A die is suspected to be biased in favour of a " 1 " occurring. A hypothesis test is carriec out. When the die is thrown 18 times, she obtained a " 1 " on 6 occasions. State appropriate hypotheses and carry out the test at the \( 10 \% \) level of significance. 6 The mayor of a certain city claims that two out of three youths living in the city are smokers A random sample of 25 youths are selected from different locations in the city. It is found that 12 of them are smokers. Does this provide sufficient evidence that the mayor has overstatec his claim at the \( 5 \% \) level of significance? 7 It is advertised that a certain type of light-bulbs have a lifespan of 1000 hours and the design is such that \( 80 \% \) meet this specification. Rajes thinks this is over optimistic. He buys 16 bulbs and records when they fail. He finds that 6 bulbs have lifespans less than as advertised and he complains to the manufacturer. Is Rajes's complaint justified at the \( 5 \% \) level of significance? 8 The manufacturer of "Supreme" margarine claims that people prefer this to butter. As par of an advertising campaign he asks 10 people to taste a sample of "Supreme" margarine and a sample of butter and say which one they prefer. 8 people choose "Supreme" margarine. Assess the manufacturer's claim in the light of this evidence at the \( 5 \% \) level of significance. 9 A biologist carries out a research to see if pollution levels affect the local ecology. A certair type of moth exists in two varieties. In one particular village, a previous piece of research shows that \( 20 \% \) of the moths are dark and the rest are light. The biologist catches a randon sample of 20 moths and counts the dark ones. At the \( 1 \% \) level of significance, how many dark ones would show that the local ecology is changing?

I. Se refieren a indicadores de posición que dividen en cuatro partes porcentuales iguales una distribución de datos ordenados. II. El segundo cuartil corresponde a la mediana de la distribución de datos. III. El tercer cuartil indica el valor mínimo que toma el \( 75 \% \) de la distribución ordena de datos. IV. El cuartil uno es equivalente al decil 2 . De las anteriores afirmaciones cuál o cuáles son verdaderas: Seleccione una: I y IV II I y II II y III.