Question

Answer all parts of question 1, each part is worth a corresponding point value.

1a On June 23, 2003, a huge, unidentified sea creature was found washed ashore on a beach in Chile. The animal was described a 40-foot long mass of rotting gray flesh that scientists estimated to weigh about 13 tons. Some scientists believed that the creature was a giant octopus, despite the absence of obvious tentacles. The largest octopi are found along the northern Pacific coast of the U.S. A team of researchers recently obtained a random sample of n = 45 out of approximately N = 8659 octopi living in the northern Pacific Ocean. The sample mean length (in feet) of each specimen was 13.47 feet with a sample standard deviation of 5.98 feet . Assume the distribution of octopus lengths is normal. Find a 99% confidence interval for the true population mean length of octopi living in the northern Pacific Ocean.

1. Determine the prerequisites have been met (3 pts):

• Random sample:
• n ≤ 0.05 of N:
• the population is normally distributed or n ≤ 30:

2. Construct the confidence interval (4 pts):

3. Interpret the confidence interval, in a statement (1 pt):

1b The Earth is structured in layers: crust, mantle, and core. A recent study was conducted to estimate the mean depth of the upper mantle in a specific farming region in California. Twenty-six, n = 26 sample sites were selected at random from a normally distributed population of approximately N = 1598 sites, and the depth of the upper mantle was measured using changes in seismic velocity and density. The sample mean was 127.5 km and the sample standard deviation was 21.3 km. Suppose the depth of the upper mantle is normally distributed. Find a 90% confidence interval for the true mean depth of the upper mantle in this farming region.

1. Determine the prerequisites have been met (3 pts):

• Random sample:
• n ≤ 0.05 of N:
• the population is normally distributed or n ≤ 30:

2. Construct the confidence interval (4 pts):

3. Interpret the confidence interval, in a statement (1 pt):

1c According to the U.S. Fire Administration, approximately N = 25,000 fires are caused by fire-works each year in the United States. Despite numerous public warnings against the use of fireworks, the home property damage due to these fires is enormous. In a random sample of n = 25 fires due to fireworks, the resulting mean property damage (in dollars) was 860.75 with a standard deviation of 350.50. Assume the underlying distribution of property damage due to these fires is normal. Find a 99% confidence interval for the true mean property damage due to a fire caused by fireworks.

1. Determine the prerequisites have been met (3 pts):

• Random sample:
• n ≤ 0.05 of N:
• the population is normally distributed or n ≤ 30:

2. Construct the confidence interval (4 pts):

3. Interpret the confidence interval, in a statement (1 pt):

1d The ambient temperature in which humans are comfortable varies with culture, activity, metabolic rate, psychological state, environment and season. For most people in the U.S., the comfort zone is 68 to 78 degrees Fahrenheit. During a recent winter, a random sample of n = 5067 homeowners was selected from the Southern region of the U.S, out of a normally distributed population of approximately N = 32,141,882 homeowners. The thermostat temperature setting (in degrees Fahrenheit) was recorded for each home. The sample mean temperature was 72.1 degrees Fahrenheit, with a standard deviation of 1.55 degrees Fahrenheit. Find a 95% confidence interval for the population mean thermostat setting in degrees Fahrenheit for homeowners from the Southern region of the U.S, during winter.

1. Determine the prerequisites have been met (3 pts):

• Random sample:
• n ≤ 0.05 of N:
• the population is normally distributed or n ≤ 30:

2. Construct the confidence interval (4 pts):

3. Interpret the confidence interval, in a statement (1 pt):

Answer 1

1(a)

• Random sample: It is given that sample is simple random sample.
• n ≤ 0.05 of N: Since n / N = 0.005 (approx) so it is fulfilled.
• the population is normally distributed or n ≤ 30: n is greater than 30 so normality can be assumed.

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(b)

(c)

We are 99% confident that true population mean length of octopi living in the northern Pacific Ocean lies in the above interval.