2. Example: Bluethroat Song Duration Male bluethroats have a complex song which is thought to be used to attract female birds. Let x denote the duration of a randomly selected song (in seconds) from a male bluethroat. The authors of research on bluethroat song report the mean song duration is 13.8 seconds and the standard deviation of song durations is 11.8 seconds. The authors also noted that the song length distribution is not normal. a. Let x = average song duration (in seconds) for a sample of 36 male bluethroat songs. Is this distribution of the sample mean song duration "normally distributed"? Explain how you know. b. Find the probability that a sample of 36 male bluethroat songs will have a mean duration greater than 12 seconds. Draw, label, and shade a graph to illustrate your result. X = average song duration (in seconds) for the sample of 36 male bluethroat songs. ~is distributed according to a distribution with = = c. Draw and label the distribution of x for this problem: d. Find P(x > 12) = e. Find P(x 16)= 3. Reach Distance. The overhead reach distances of adult women are normally distributed with a mean of 200 cm. and a standard deviation of 12.5 cm. (Illustrate each with a graph!) a) Find the probability that an individual distance is greater than 205 cm. random variable: distribution: b) c) Find the probability that the mean reach distance for 25 randomly selected adult women is greater than 205 cm. random variable: distribution: Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? SECTION 7.4 SAMPLING DISTRIBUTION OF P THE CENTRAL LIMIT THEOREM FOR PROPORTIONS: Let p be the sample proportion for a sample size of n and population proportion p. if np 10 and n(1 - p) 10 then the distribution of p is approximately normal, with mean and standard deviation = p_and_ = p(1 - p) n 1. The soft-drink cups at a certain fast-food restaurant have tickets attached to them. Customers peel off the tickets to see whether they win a prize. The proportion of tickets that are winners is p = 0.25. A total of n = 70 people purchase soft drinks between noon and 1:00 PM on a certain day. Let p be the proportion that win a prize. a. Is it appropriate to use the normal distribution to find probabilities for p? b. Find the mean and standard deviation of p. 2. According to a Harris poll taken in September 2013, chocolate is the favorite ice cream flavor for 27% of Americans. If a sample of 100 Americans is taken, a. Is it appropriate to use the normal distribution to find probabilities for p? b. what is the probability that the sample proportion of those who prefer chocolate is greater than 0.30? 3. The General Social Survey reported that 56% of American adults saw a doctor for an illness during the past year. A sample of 65 adults is drawn. a. Is it appropriate to use the normal distribution to find probabilities for p? b. What is the probability that more than 60% of them saw a doctor? c. Would it be unusual if more than 70% of them saw a doctor?