Иностранная литература и современность: General Statistics: Ch 6 HW Flashcards

What requirements are necessary for a normal probability distribution to be a standard normal probability​ distribution?

The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 8 minutes. Find the probability that a randomly selected passenger has a waiting time less than 0.75 minutes.

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.

Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.

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 and draw a sketch of the region.

Assume the readings on thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C.

Find the probability that a randomly selected thermometer reads greater than –1.78 and draw a sketch of the region.

Assume the readings on thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C.

Find the probability that a randomly selected thermometer reads between –1.03 and –0.02 and draw a sketch of the region.

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 between –2.02 and 3.84 and draw a sketch of the region.

Assume that the readings on the thermometers are normally distributed with a mean of 0° and standard deviation of 1.00°C. A thermometer is randomly selected and tested.

Draw a sketch and find the temperature reading corresponding to P₉₇​, the 97th percentile. This is the temperature reading separating the bottom 97% from the top 3%.

Assume that the readings on the thermometers are normally distributed with a mean of 0° and standard deviation of 1.00°C. Assume 2.1​% of the thermometers are rejected because they have readings that are too high and another 2.1​% are rejected because they have readings that are too low.

Draw a sketch and find the two readings that are cutoff values separating the rejected thermometers from the others.

Find the indicated area under the curve of the standard normal​ distribution, then convert it to a percentage and fill in the blank.

Which of the following groups of terms can be used interchangeably when working with normal​ distributions?

Which of the following is NOT a requirement for a density​ curve?

Which of the following does NOT describe the standard normal​ distribution?

Which of the following would be information in a question asking you to find the area of a region under the standard normal curve as a​ solution?

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 3.61 and draw a sketch of the region.

Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1.

Draw a graph and find P₃, the 3rd percentile. This is the bone density score separating the bottom 3% from the top 97%.

Find the indicated critical value.

z_0.04

A survey found that​ women's heights are normally distributed with mean 63.6 in and standard deviation 2.3 in. A branch of the military requires​ women's heights to be between 58 in and 80 in.

1. Find the percentage of women meeting the height requirement. Are many women being denied the opportunity to join this branch of the military because they are too short or too​ tall?
2. If this branch of the military changes the height requirements so that all women are eligible except the shortest​ 1% and the tallest​ 2%, what are the new height​ requirements?

Men's heights are normally distributed with mean 68.7 in and standard deviation of 2.8 in. ​Women's heights are normally distributed with mean 63.5 in and standard deviation of 2.5 in. The standard doorway height is 80 in.

1. What percentage of men are too tall to fit through a standard doorway without​ bending, and what percentage of women are too tall to fit through a standard doorway without​ bending?
2. If a statistician designs a house so that all of the doorways have heights that are sufficient for all men except the tallest​ 5%, what doorway height would be​ used?

The lengths of pregnancies are normally distributed with a mean of 266 days and a standard deviation of 15 days.

1. Find the probability of a pregnancy lasting 307 days or longer.
2. If the length of pregnancy is in the lowest 2​%, then the baby is premature. Find the length that separates premature babies from those who are not premature.

Assume that the Richter scale magnitudes of earthquakes are normally distributed with a mean of 1.087 and a standard deviation of 0.562.

1. Earthquakes with magnitudes less than 2.000 are considered​ "microearthquakes" that are not felt. What percentage of earthquakes fall into this​ category?
2. Earthquakes above 4.0 will cause indoor items to shake. What percentage of earthquakes fall into this​ category?
3. Find the 95th percentile.
4. Will all earthquakes above the 95th percentile cause indoor items to​ shake?

Chocolate chip cookies have a distribution that is approximately normal with a mean of 23.1 chocolate chips per cookie and a standard deviation of 2.6 chocolate chips per cookie.

Find P₅ and P₉₅.

How might those values be helpful to the producer of the chocolate chip​ cookies?

Pulse rates of women are normally distributed with a mean of 77.5 beats per minute and a standard deviation of 11.6 beats per minute.

What are the values of the mean and standard deviation after converting all pulse rates of women to z scores using z = (x – ​µ) ÷ σ?

The original pulse rates are measure with units of​ "beats per​ minute". What are the units of the corresponding z​ scores?

Find the area of the shaded region. The graph to the right depicts IQ scores of​ adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.

Assume that adults have IQ scores that are normally distributed with a mean of µ = 100 and a standard deviation σ = 15.

Find the probability that a randomly selected adult has an IQ less than 118.

Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation 15. Find P₃​, which is the IQ score separating the bottom 3​% from the top 97​%.

Find the area of the shaded region. The graph to the right depicts IQ scores of​ adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.

Find the indicated IQ score. The graph to the right depicts IQ scores of​ adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.

Which of the following is not​ true?

1. A​ z-score is an area under the normal curve.
2. A​ z-score is a conversion that standardizes any value from a normal distribution to a standard normal distribution.
3. The area in any normal distribution bounded by some score x is the same as the area bounded by the equivalent​ z-score in the standard normal distribution.
4. If values are converted to standard​ z-scores, then procedures for working with all normal distributions are the same as those for the standard normal distribution.

What conditions would produce a negative​ z-score?

Which of the following statistics are unbiased estimators of population​ parameters?

1. Sample proportion used to estimate a population proportion.
2. Sample range used to estimate a population range.
3. Sample variance used to estimate a population variance.
4. Sample mean used to estimate a population mean.
5. Sample standard deviation used to estimate a population standard deviation.
6. Sample median used to estimate a population median.

The assets​ (in billions of​ dollars) of the four wealthiest people in a particular country are 37, 27, 22, 19. Assume that samples of size n = 2 are randomly selected with replacement from this population of four values.

1. After identifying the 16 different possible samples and finding the mean of each​ sample, construct a table representing the sampling distribution of the sample mean. In the​ table, values of the sample mean that are the same have been combined.
2. Compare the mean of the population to the mean of the sampling distribution of the sample mean.
3. Do the sample means target the value of the population​ mean? In​ general, do sample means make good estimates of population​ means? Why or why​ not?

Which of the following is NOT a property of the sampling distribution of the sample​ mean?

1. The sample means target the value of the population mean.
2. The distribution of the sample mean tends to be skewed to the right or left.
3. The mean of the sample means is the population mean.
4. The expected value of the sample mean is equal to the population mean.

Which of the following is a biased​ estimator? That​ is, which of the following does not target the population​ parameter?

The population of current statistics students has ages with mean µ and standard deviation σ. Samples of statistics students are randomly selected so that there are exactly 37 students in each sample. For each​ sample, the mean age is computed.

What does the central limit theorem tell us about the distribution of those mean​ ages?

Which of the following is NOT a conclusion of the Central Limit​ Theorem?

1. The standard deviation of all sample means is the population standard deviation divided by the square root of the sample size.
2. The mean of all sample means is the population mean µ.
3. The distribution of the sample means x̄ ​will, as the sample size​ increases, approach a normal distribution.
4. The distribution of the sample data will approach a normal distribution as the sample size increases.

Which of the following is not a commonly used​ practice?

1. If the distribution of the sample means is normally​ distributed, and n > 30, then the population distribution is normally distributed.
2. The distribution of sample means gets closer to a normal distribution as the sample size n gets larger.
3. If the original population is not normally distributed and n > 30, the distribution of the sample means can be approximated reasonably well by a normal distribution.
4. If the original population is normally​ distributed, then for any sample size​ n, the sample means will be normally distributed.

Assume that​ women's heights are normally distributed with a mean given by µ = 63.6 in​, and a standard deviation given by σ = 2.6 in.

(a) If 1 woman is randomly​ selected, find the probability that her height is less than 64 in.

(b) If 44 women are randomly​ selected, find the probability that they have a mean height less than 64 in.

Assume that​ women's heights are normally distributed with a mean given by µ = 62.2 in​, and a standard deviation given by σ = 2.8 in.

1. If 1 woman is randomly​ selected, find the probability that her height is between 62.0 in and 63.0 in.
2. If 15 women are randomly​ selected, find the probability that they have a mean height between 62.0 in and 63.0 in.

Women have head circumferences that are normally distributed with a mean given by µ = 24.27 in​., and a standard deviation given by σ = 0.8 in.

1. If a hat company produces​ women's hats so that they fit head circumferences between 23.6 in. and 24.6 ​in., what is the probability that a randomly selected woman will be able to fit into one of these​ hats?
2. If the company wants to produce hats to fit all women except for those with the smallest 1.25​% and the largest 1.25​% head​ circumferences, what head circumferences should be​ accommodated?
3. If 12 women are randomly​ selected, what is the probability that their mean head circumference is between 23.6 in. and 24.6 ​in.?
4. If this probability is​ high, does it suggest that an order of 12 hats will very likely fit each of 12 randomly selected​ women? Why or why​ not? (Assume that the hat company produces​ women's hats so that they fit head circumferences between 23.6 in. and 24.6 ​in.)

A ski gondola carries skiers to the top of a mountain. It bears a plaque stating that the maximum capacity is 12 people or 1920 lb. That capacity will be exceeded if 12 people have weights with a mean greater than (1920 lb)/12 = 160 lb. Assume that weights of passengers are normally distributed with a mean of 177.5 lb and a standard deviation of 40.4 lb.

1. Find the probability that if an individual passenger is randomly​ selected, their weight will be greater than 160 lb.
2. Find the probability that 12 randomly selected passengers will have a mean weight that is greater than 160 lb​ (so that their total weight is greater than the gondola maximum capacity of 1920 ​lb)

An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 150 lb and 191 lb. The new population of pilots has normally distributed weights with a mean of 156 lb and a standard deviation of 25.7 lb.

1. If a pilot is randomly​ selected, find the probability that his weight is between 150 lb and 191 lb.
2. If 33 different pilots are randomly​ selected, find the probability that their mean weight is between 150 lb and 191 lb.
3. When redesigning the ejection​ seat, which probability is more​ relevant?

An airliner carries 200 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 69.0 in and a standard deviation of 2.8 in.

1. If a male passenger is randomly​ selected, find the probability that he can fit through the doorway without bending.
2. If half of the 200 passengers are​ men, find the probability that the mean height of the 100 men is less than 72 in.
3. When considering the comfort and safety of​ passengers, which result is more​ relevant: the probability from part​ (a) or the probability from part​ (b)? Why?
4. When considering the comfort and safety of​ passengers, why are women ignored in this​ case?

Before every​ flight, the pilot must verify that the total weight of the load is less than the maximum allowable load for the aircraft. The aircraft can carry 35 ​passengers, and a flight has fuel and baggage that allows for a total passenger load of 5,880 lb. The pilot sees that the plane is full and all passengers are men. The aircraft will be overloaded if the mean weight of the passengers is greater than (5,880 lb)/35 = 168 lb.

What is the probability that the aircraft is​ overloaded?

Should the pilot take any action to correct for an overloaded​ aircraft? Assume that weights of men are normally distributed with a mean of 175.5 lb and a standard deviation of 38.7.

The value given below is discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability.

Probability of exactly 5 passengers who do not show up for a flight

If np ≥ 5 and nq ≥ 5, estimate P(more than 5) with n = 12 and p = 0.4 by using the normal distribution as an approximation to the binomial​ distribution; if np < 5 or nq < 5, then state that the normal approximation is not suitable.

If np ≥ 5 and nq ≥ 5, estimate P(more than 5) with n = 13 and p = 0.6 by using the normal distribution as an approximation to the binomial​ distribution; if np < 5 or nq < 5, then state that the normal approximation is not suitable.

A scientist conducted a hybridization experiment using peas with green pods and yellow pods. He crossed peas in such a way that​ 25% (or 144​) of the 576 offspring peas were expected to have yellow pods. Instead of getting 144 peas with yellow​ pods, he obtained 149. Assume that the rate of​ 25% is correct.

1. Find the probability that among the 576 offspring​ peas, exactly 149 have yellow pods.
2. Find the probability that among the 576 offspring​ peas, at least 149 have yellow pods.
3. Which result is useful for determining whether the claimed rate of​ 25% is​ incorrect? (Part​ (a) or part​ (b)?)
4. Is there strong evidence to suggest that the rate of​ 25% is​ incorrect?

A​ gender-selection technique is designed to increase the likelihood that a baby will be a girl. In the results of the​ gender-selection technique, 809 births consisted of 422 baby girls and 387 baby boys. In analyzing these​ results, assume that boys and girls are equally likely.

1. Find the probability of getting exactly 422 girls in 809 births.
2. Find the probability of getting 422 or more girls in 809 births. If boys and girls are equally​ likely, is 422 girls in 809 births unusually​ high?
3. Which probability is relevant for trying to determine whether the technique is​ effective: the result from part​ (a) or the result from part​ (b)?
4. Based on the​ results, does it appear that the​ gender-selection technique is​ effective?

In a survey of 1227 ​people, 828 people said they voted in a recent presidential election. Voting records show that 65​% of eligible voters actually did vote. Given that 65​% of eligible voters actually did​ vote.

1. Find the probability that among 1227 randomly selected​ voters, at least 828 actually did vote.
2. What do the results from part​ (a) suggest?

Based on a recent​ survey, 20​% of adults in a specific country smoke. In a survey of 100 ​students, it is found that 18 of them smoke.

Find the probability that should be used for determining whether the 20​% rate is correct for students.

What can be​ concluded?

Which of the following is NOT a requirement for using the normal distribution as an approximation to the binomial​ distribution?

1. The sample is a simple random sample of size n from a population in which the proportion of successes is​ p, or the sample is the result of conducting n independent trials of a binomial experiment in which the probability of success is p.
2. nq ≥ 5
3. The sample is the result of conducting several dependent trials of an experiment in which the probability of success is p.
4. np ≥ 5

Which statement below indicates the area to the left of 19.5 before a continuity correction is​ used?

1. At least 19
2. At least 20
3. Less than 19
4. At most 19

A continuity correction is made to a discrete whole number x in the binomial distribution by representing the discrete whole number x by which of the following​ intervals?