1. Assume that you have applied for two scholarships, a Leadership scholarship (denoted by L) and a Merit scholarship (denoted by M). The probability that you receive a Leadership scholarship is 0.25. The probability of receiving both scholarships is 0.15. The probability of receiving at least one of the scholarships is 0.45.

Write the probability statement using the events defined in the problem, e.g., P(L), P(M). Then compute the probability. Use 4-decimal accuracy when necessary. Please show steps.

a) [3 points] The probability that you will receive a Merit scholarship is: P(__________) = __________. Show detailed steps below.

b) [3 points] Events L and M __________ (Choose one: are, are not) mutually exclusive. Why or why not? Explain below.

c) [3 points] Events L and M __________ (Choose one: are, are not) independent. Why or why not? Explain below.

d) [3 points] The probability of receiving the Leadership scholarship given that you have been awarded the Merit scholarship is P(__________) = __________. Show detailed steps below.

e) [3 points] The probability of receiving the Merit scholarship given that you have been awarded the Leadership scholarship is: P(__________) = __________. Show detailed steps below.

1. Suppose the heights of 18-year-old men are approximately normally distributed, with mean 71 inches and standard deviation 5 inches.

(a) What is the probability that an 18-year-old man selected at random is between 70 and 72 inches tall? (Round your answer to four decimal places.)


(b) If a random sample of twenty-three 18-year-old men is selected, what is the probability that the mean height x is between 70 and 72 inches? (Round your answer to four decimal places.)


(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

The probability in part (b) is much higher because the mean is larger for the x distribution.The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.     The probability in part (b) is much higher because the mean is smaller for the x distribution.The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.The probability in part (b) is much higher because the standard deviation is larger for the x distribution.

A manufacturing company regularly conducts quality control checks at specified periods on the products it manufactures.​ Historically, the failure rate for LED light bulbs that the company manufactures is 18​%. Suppose a random sample of 10 LED light bulbs is selected. Complete parts​ (a) through​ (d) below. a. What is the probability that none of the LED light bulbs are​ defective? The probability that none of the LED light bulbs are defective is . 1374. ​(Type an integer or a decimal. Round to four decimal places as​ needed.) b. What is the probability that exactly one of the LED light bulbs is​ defective? The probability that exactly one of the LED light bulbs is defective is . 3017. ​(Type an integer or a decimal. Round to four decimal places as​ needed.) c. What is the probability that four or fewer of the LED light bulbs are​ defective? The probability that four or fewer of the LED light bulbs are defective is nothing. ​(Type an integer or a decimal. Round to four decimal places as​ needed.) d. What is the probability that five or more of the LED light bulbs are​ defective? The probability that five or more of the LED light bulbs are defective is nothing. ​(Type an integer or a decimal. Round to four decimal places as​ needed.)

Suppose the heights of 18-year-old men are approximately normally distributed, with mean 66 inches and standard deviation 6 inches.

(a) What is the probability that an 18-year-old man selected at random is between 65 and 67 inches tall? (Round your answer to four decimal places.)


(b) If a random sample of twenty-three 18-year-old men is selected, what is the probability that the mean height x is between 65 and 67 inches? (Round your answer to four decimal places.)


(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.

The probability in part (b) is much higher because the mean is smaller for the x distribution.     

The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.

The probability in part (b) is much higher because the standard deviation is larger for the x distribution.

The probability in part (b) is much higher because the mean is larger for the x distribution.

Investment advisors agree that​ near-retirees, defined as people aged 55 to​ 65, should have balanced portfolios. Most advisors suggest that the​ near-retirees have no more than​ 50% of their investments in stocks.​ However, during the huge decline in the stock market in​ 2008, 23​% of​ near-retirees had 85​% or more of their investments in stocks. Suppose you have a random sample of 10 people who would have been labeled as​ near-retirees in 2008. Complete parts​ (a) through​ (d) below.

a. What is the probability that during 2008 none had 85​% or more of their investment in​ stocks? The probability is . ​(Round to four decimal places as​ needed.)

b. What is the probability that during 2008 exactly one had 85​% or more of his or her investment in​ stocks? The probability is . ​(Round to four decimal places as​ needed.)

c. What is the probability that during 2008 two or fewer had 85​% or more of their investment in​ stocks? The probability is . ​(Round to four decimal places as​ needed.)

d. What is the probability that during 2008 three or more had 85​% or more of their investment in​ stocks? The probability is . ​(Round to four decimal places as​ needed.)

Suppose the heights of 18-year-old men are approximately normally distributed, with mean 68 inches and standard deviation 2 inches.

(a) What is the probability that an 18-year-old man selected at random is between 67 and 69 inches tall? (Round your answer to four decimal places.)


(b) If a random sample of twenty-nine 18-year-old men is selected, what is the probability that the mean height x is between 67 and 69 inches? (Round your answer to four decimal places.)


(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

The probability in part (b) is much higher because the mean is larger for the x distribution.The probability in part (b) is much higher because the standard deviation is larger for the x distribution.    The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.The probability in part (b) is much higher because the mean is smaller for the x distribution.The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.

Suppose the heights of 18-year-old men are approximately normally distributed, with mean 68 inches and standard deviation 4 inches.

(a) What is the probability that an 18-year-old man selected at random is between 67 and 69 inches tall? (Round your answer to four decimal places.)


(b) If a random sample of twenty-six 18-year-old men is selected, what is the probability that the mean height x is between 67 and 69 inches? (Round your answer to four decimal places.)


(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

1The probability in part (b) is much higher because the standard deviation is smaller for the x distribution. 2The probability in part (b) is much higher because the mean is larger for the x distribution. 3The probability in part (b) is much lower because the standard deviation is smaller for the x distribution. 4The probability in part (b) is much higher because the standard deviation is larger for the x distribution.5The probability in part (b) is much higher because the mean is smaller for the x distribution.

6.1

16. ASSUME THAT A RANDOMLY SELECTED SUBJECT IS GIVEN A BONE DENSITY TEST. THOSE TEST SCORES ARE NORMALLY DISTRIBUTED WITH A MEAN OF 0 AND A STANDARD DEVIATION OF 1. FIND THE PROBABILITY THAT A GIVEN SCORE IS LESS THAN 1.66. THE PROBABILITY IS.......(ROUND TO 4 DECIMAL PLACES AS NEEDED)

17.ASSUME THAT A RANDOMLY SELECTED SUBJECT IS GIVEN A BONE DENSITY TEST. THOSE TEST SCORES ARE NORMALLY DISTRIBUTED WITH A MEAN OF 0 AND A STANDARD DEVIATION OF 1. FIND THE PROBABILITY OF A BONE DENSITY TEST SCORE GREATER THAN 0.66. THE PROBABILITY IS.......(ROUND TO 4 DECIMAL PLACES AS NEEDED)

18. ASSUME THAT A RANDOMLY SELECTED SUBJECT IS GIVEN A BONE DENSITY TEST. THOSE TEST SCORES ARE NORMALLY DISTRIBUTED WITH A MEAN OF 0 AND A STANDARD DEVIATION OF 1. FIND THE PROBABILITY OF A BONE DENSITY TEST SCORE GREATER THAN -1.98. THE PROBABILITY IS.......(ROUND TO 4 DECIMAL PLACES AS NEEDED)

19.ASSUME THAT A RANDOMLY SELECTED SUBJECT IS GIVEN A BONE DENSITY TEST. THOSE TEST SCORES ARE NORMALLY DISTRIBUTED WITH A MEAN OF 0 AND A STANDARD DEVIATION OF 1. FIND THE PROBABILITY OF A BONE DENSITY TEST SCORE BETWEEN -1.81 AND 1.81. THE PROBABILITY IS.......(ROUND TO 4 DECIMAL PLACES AS NEEDED)