sta midterm ch6
Assume that the readings on the thermometers are normally distributed with a mean of 0° and standard deviation of 1.00°C. Assume 3.3% of the thermometers are rejected because they have readings that are too high and another 3.3% 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.
-1.83, 1.84 1-.033 =.967 .033 look on z score sheet for area of .967, that z score will be for the higher value, look on z score sheet for .033, that z score will be for the lower value
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. graph shows between the z scores of 70 and 125
.9297 z= 70 - 100/15 z= -30/15 z=-2 look up -2 in the z score chart =.0228 z=125-100/15 z=25/15 z=1.67 look up 1.67 in the z score chart =.9525 .9525-.0228 =.9297
A statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed between 46.0 and 56.0 minutes. Find the probability that a given class period runs between 51.5 and 51.75 minutes.
0.025 56-46= 10 height- .1 1=10(x) P(between 51.5 and 51.75)= (shaded region length)(shaded region height) =(51.75-51.5)(.1) =0.025
Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. z=.14
0.5557 look at the z score table and find the z score of 0.14
Where would a value separating the top 15% from the other values on the graph of a normal distribution be found?
the right side of the horizontal scale of the graph
Which is NOT a criterion for distinguishing between results that could easily occur by chance and those results that are highly unusual?
the sample size is less than 5% of the size of the population
_____________ is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population.
the sampling distribution of a statistic
_____________ is the distribution of sample proportions, with all samples having the same sample size n taken from the same population.
the sampling distribution of the proportion
A continuous random variable has a _______ distribution if its values are spread evenly over the range of possibilities.
uniform
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?
x−0.5 to x+0.5
The notation P(z<a) denotes _______.
the probability that the z score is less than a
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. graph has a z score of 105
0.6293 z = x-mean/standard deviation z = 105-100/15 z = .3333 look up .33 in z score chart 0.6293
The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 5 minutes. Find the probability that a randomly selected passenger has a waiting time less than 3.75 minutes.
0.75 the total area under a distribution curve is 1. Find the length of the distribution curve, 5-0, 5. 1=5(x) x=0.2 height P(less than 3.75) = (length of shaded region)(height of shaded region) =(3.75-0)(.2) =(3.75)(.2) =.75
Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. Area=.9382
1.54 Use area and look for number closest to area on z score chart and find corresponding z score.
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 Psubscript96 or 96th percentile. This is the temperature reading separating the bottom 96% from the top 4%.
1.75 There is an area of .96. Look for the closest to .96 in the z score table and find the corresponding z score.
Find the indicated critical value. Z subscript 0.03
1.88 1-.033 .997 find corresponding z score according to the area .997
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. Graph shows an area of .97
128.2 look up .97 in the z score chart 1.88, set it equal to the z score equation x-100/15 = 1.88 x-100=28.2 x=128.2
Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation 20. Find P subscript 10, which is the IQ score separating the bottom 10% from the top 90%.
74.4 look up .1 and .9 in the z score chart -1.28 and 1.28 1.28 = x-100/20 125.6 = x -1.28 = x-100/20 74.4 = x
Find the indicated area under the curve of the standard normal distribution, then convert it to a percentage and fill in the blank. About _____% of the area is between z= -1.7 and z=1.7 (or within 1.7 standard deviations of the mean).
91.08% look on z score chart, subtract area of pos z score - area of negative z score .9554-.0446= .9108 or 91.08%
Which of the following is not a commonly used practice?
If the distribution of the sample means is normally distributed, and n>30, then the population distribution is normally distributed.
Which of the following is not true?
A z-score is an area under the normal curve.
The population of current statistics students has ages with mean muμ and standard deviation sigmaσ. Samples of statistics students are randomly selected so that there are exactly 59 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?
Because n>30, the sampling distribution of the mean ages can be approximated by a normal distribution with μ and σ/square root of 59
Which of the following is NOT a conclusion of the Central Limit Theorem?
The distribution of the sample data will approach a normal distribution as the sample size increases.
Which of the following is NOT a descriptor of a normal distribution of a random variable?
The graph is centered around 0
Why must a continuity correction be used when using the normal approximation for the binomial distribution?
The normal distribution is a continuous probability distribution being used as an approximation to the binomial distribution which is a discrete probability distribution.
Which of the following is NOT a requirement for using the normal distribution as an approximation to the binomial distribution?
The sample is the result of conducting several dependent trials of an experiment in which the probability of success is p.
If you are asked to find the 85th percentile, you are being asked to find _____.
a data value associated with an area of .85 to its left
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?
a distance on the horizontal axis is given
What conditions would produce a negative z-score?
a z score corresponding to an area located entirely in the left side of the curve
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 fewer than 3 passengers who do not show up for a flight
the area to the left of 2.5
Which of the following is NOT a property of the sampling distribution of the sample mean?
the distribution of a sample mean tends to be skewed to the right or left
The probability of flu symptoms for a person not receiving any treatment is 0.039. In a clinical trial of a common drug used to lower cholesterol, 41 of 964 people treated experienced flu symptoms. Assuming the drug has no effect on the likelihood of flu symptoms, estimate the probability that at least 41 people experience flu symptoms. What do these results suggest about flu symptoms as an adverse reaction to the drug? (a) P(X≥41)= (b) What does the result from part (a) suggest?
a. .3156 np=(964)(.039) =37.596 nq=(964)(.961) =926.404 μ=np=37.596 σ=square root of npq=6.010803274 p(x≥41) as 1-p(x<41) x<41-.5=40.5 40.5-37.596/6.010803274 =.48=.6844 1-.6844=.3156 b. The drug has no effect on flu symptoms because x≥41 is not highly unlikely.
Women have head circumferences that are normally distributed with a μ=22.31 in., and a σ=0.6 in. a. If a hat company produces women's hats so that they fit head circumferences between 21.7 in. and 22.7 in., what is the probability that a randomly selected woman will be able to fit into one of these hats? b. 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? c. If 12 women are randomly selected, what is the probability that their mean head circumference is between 21.7 in. and 22.7 in.? 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 21.7 in. and 22.7 in.)
a. .5883 21.7-22.31/.6= -1.02 =.1539 22.7-22.31/.6= .65 =.7422 .7422-.1539= .5883 b. 20.97 and 23.65 .0125 and .9875 =-2.24 and 2.24 -2.24=x-22.31/.6 =20.97 2.24=x-22.32/.6 =23.65 c. .9876, No, the hats must fit individual women, not the mean from 12 women. If all hats are made to fit head circumferences between 21.7 in. and 22.7 in., the hats won't fit about half of those women. .6/square root of 12=.1732050808 21.7-22.31/.1732050808= -3.52 =.0002 22.7-22.31/.1732050808= 2.25 =.9878 .9878-.0002= .9876
Assume that women's heights are normally distributed with a μ=62.2 in, and a σ=2.8 in. (a) If 1 woman is randomly selected, find the probability that her height is less than 63 in. (b) If 44 women are randomly selected, find the probability that they have a mean height less than 63 in.
a. .6141 63-62.2/2.8 =.29 find corresponding area on z score chart b. .9713 =σ/square root of n 2.8/square root of 44= .4221158824 63-62.2/.4221158824 = z score of 1.9 =.9713
A survey found that women's heights are normally distributed with mean 63.5 in and standard deviation 2.5 in. A branch of the military requires women's heights to be between 58 in and 80 in. a. 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? b. 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?
a. .986 or 98.6% =58-63.5/2.5 =-2.2 =.0139 =80-63.5/2.5 =6.6 =.9999 .9999-.0139= .986 b. 57.7 and 68.6 .01 and .98 -2.33 and 2.05 -2.33 = x-63.5/2.5 x= 57.675 2.05=x-63.5/2.5 x= 68.625
A scientist conducted a hybridization experiment using peas with green pods and yellow pods. He crossed peas in such a way that 25% (or 138) of the 552 offspring peas were expected to have yellow pods. Instead of getting 138 peas with yellow pods, he obtained 144. Assume that the rate of 25% is correct. a. Find the probability that among the 552 offspring peas, exactly 144 have yellow pods. b. Find the probability that among the 552 offspring peas, at least 144 have yellow pods. c. Which result is useful for determining whether the claimed rate of 25% is incorrect? (Part (a) or part (b)?) d. Is there strong evidence to suggest that the rate of 25% is incorrect?
a. 0.0335 nq=552(.75)=414 nq>=5 so good μ=np =552(.25)=138 σ=square root of npq =square root of 552(.25)(.75) =10.17349497 x=144 x-.5=143.5 x+.5=144.5 z=143.5-138/10.173349497 =.5406=.54 z=144.5-138/10.173349497 =.6389=.64 .7389-.7054=.0335 b. .2946 x=143.5 z=,54 area=.7054 1-.7054= .2946 c. part b is more useful d. no
The overhead reach distances of adult females are normally distributed with a mean of 195 cm and a standard deviation of 8 cm. a. Find the probability that an individual distance is greater than 208.40 cm. b. Find the probability that the mean for 20 randomly selected distances is greater than 192.80 cm. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?
a. 0.0465 208.4-195/8= 1.675 =.9535 =1-.9535 =.0465 b. 0.8906 8/square root of 20= 1.788854382 192.8-195/1.788854382= z score of -1.23 =.1094 1-.1094 =.8906 c. The normal distribution can be used because the original population has a normal distribution.
Which of the following is NOT a requirement for a density curve?
the graph is centered around 0
Which of the following does NOT describe the standard normal distribution?
the graph is uniform
The assets (in billions of dollars) of the four wealthiest people in a particular country are 38, 35, 19, 12. Assume that samples of size n=2 are randomly selected with replacement from this population of four values. a. 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. b. Compare the mean of the population to the mean of the sampling distribution of the sample mean. c. 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?
a. Sample Mean: 38, 36.5, 35, 28.5, 27, 25, 23.5, 19, 15.5, 12 Sample Probability: 1/16, 2/16, 1/16, 2/16, 2/16, 2/16, 2/16, 1/16, 2/16, 1/16 Put together all of the different combos of 38, 35, 19, 12 into pairs. Example: 12,12 12,19 12,35 12,38 19,12 19,19 19,35 19,38...etc... Find the means of each pair. Use mode to find the probability out of all of the pairs. Add up sample means and find the mean of those. mean = 26 b. The mean of the population, 26, is equal to the mean of the sample means, 26 Mean of 38, 35, 19, 12 = 26 = 26 c. The sample means target the population mean. In general, sample means do make good estimates of population means because the mean is an unbiased estimator.
Three randomly selected households are surveyed. The numbers of people in the households are 2, 3, and 10. Assume that samples of size n=2 are randomly selected with replacement from the population of 2, 3, and 10. Construct a probability distribution table that describes the sampling distribution of the proportion of even numbers when samples of sizes n=2 are randomly selected. a. Does the mean of the sample proportions equal the proportion of even numbers in the population? b. Do the sample proportions target the value of the population proportion? c. Does the sample proportion make a good estimator of the population proportion? Listed below are the nine possible samples. 2,2 2,3 2,10 3,2 3,3 3,10 10,2 10,3 10,10
a. Sample Proportion: 1, 1/2, 1, 1/2, 0, 1/2, 1, 1/2, 1 Sample Probability: 1/9, 4/9, 4/9 Find the proportion of the pairs that have even numbers. For example 2,2 = 2/2 = 1 and 2,3 = 1/2. Use mode to find the probability out of the number of pairs. Find the mean of the sample proportions, 2/3 b. The proportion of even numbers in the population is equal to the mean of the sample proportions. 2, 3, 10 = 2/3 = 2/3 c. The sample proportions target the proportion of even numbers in the population, so sample proportions make good estimators of the population proportion.
Three randomly selected households are surveyed. The numbers of people in the households are 1, 4, and 10. Assume that samples of size n=2 are randomly selected with replacement from the population of 1, 4, and 10. Listed below are the nine different samples. Complete parts (a) through (c). 1,1 1,4 1,10 4,1 4,4 4,10 10,1 10,4 10,10 a. Find the median of each of the nine samples, then summarize the sampling distribution of the medians in the format of a table representing the probability distribution of the distinct median values. b. Compare the population median to the mean of the sample medians. Choose the correct answer below. c. Do the sample medians target the value of the population median? In general, do sample medians make good estimators of population medians? Why or why not?
a. Sample median: 1, 2.5, 4, 5.5, 7, 10 Probability: 1/9, 2/9, 1/9, 2/9, 2/9, 1/9 Add the two numbers in the pair together, divide by 2. Repeat for each set of pairs. Use mode to find probability out of the number of pairs. b. The population median is not equal to the mean of the sample medians (it is also not half or double the mean of the sample medians). Population numbers- 1, 4, 10 Population mean- 4 c. The sample medians do not target the population median, so sample medians do not make good estimators of population medians.
What requirements are necessary for a normal probability distribution to be a standard normal probability distribution?
the mean and standard deviation have requirements of mean = 0 and standard deviation = 1.
Which of the following groups of terms can be used interchangeably when working with normal distributions?
areas, probabilities, and relative frequencies
Which statement below indicates the area to the left of 19.5 before a continuity correction is used?
at most 19
The _______ tells us that for a population with any distribution, the distribution of the sample means approaches a normal distribution as the sample size increases.
central limit theorem
Finding probabilities associated with distributions that are standard normal distributions is equivalent to _______.
finding the area of the shaded region representing that probability
Which of the following is a biased estimator? That is, which of the following does not target the population parameter?
median
The _______ states that if, under a given assumption, the probability of a particular observed event is exceptionally small (such as less than 0.05), we conclude that the assumption is probably not correct.
rare event rule for inferential statistics
Which of the following statistics are unbiased estimators of population parameters?
sample variance used to estime a population variance sample proportion used to estimate a population proportion sample mean used to estimate a population mean
The standard deviation of the distribution of sample means is _______.
standard deviation/ square root of n
