6.4 - 6.6 Homework

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 interval is x−0.5 to x+0.5 because 0.5 needs to be added and subtracted from x to create the interval.

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

At most 19​ "At most​ 19" indicates the area to the left of 19.5 before a continuity correction is used.

Examine the normal quantile plot and determine whether it depicts sample data from a population with a normal distribution.

Does the normal quantile plot depict sample data from a population with a normal​ distribution? No. The points exhibit some systematic pattern that is not a straight-line pattern

A researcher collects a simple random sample of​ grade-point averages of statistics​ students, and she calculates the mean of this sample. Under what conditions can that sample mean be treated as a value from a population having a normal​ distribution?

If the population of​ grade-point averages has a normal distribution The sample has more than 30​ grade-point averages.

Use the given data values​ (a sample of female arm circumferences in​ centimeters) to identify the corresponding z scores that are used for a normal quantile​ plot, then identify the coordinates of each point in the normal quantile plot. Construct the normal quantile​ plot, then determine whether the data appear to be from a population with a normal distribution. 33.8​, 41.1​, 38.8​, 32.5​, 44.4

List the z scores for the normal quantile plot. −1.28​, −0.52​, 0​, 0.52​, 1.28 Identify the coordinates of each point in the normal quantile plot. Use ordered pairs for the form​ (x,y), where x is the sorted arm circumferences in ascending​ order, and y is the corresponding z score. (32.5, −1.28)​, (33.8, −0.52)​, (38.8,0)​, (41.1,0.52)​, (44.4,1.28) Do the data come from a normally distributed​ population? Yes. The pattern of the points is reasonably close to a straight line.

Annual incomes of statistics students are known to have a distribution that is skewed to the right instead of being normally distributed. Assume that we collect a random sample of annual incomes of 50 statistics students. Can the distribution of incomes in that sample be approximated by a normal distribution because the sample is​ large? Why or why​ not?

No; the sample means will be normally​ distributed, but the sample of incomes will be skewed to the right.

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 6 passengers who do not show up for a flight

The area to the left of 5.5

The normal quantile plot shown to the right represents duration times​ (in seconds) of eruptions of a certain geyser from the accompanying data set. Examine the normal quantile plot and determine whether it depicts sample data from a population with a normal distribution.

The distribution is normal. The points are reasonably close to a straight line and do not show a systematic pattern that is not a straight-line pattern

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. The Central Limit Theorem applies to the sampling distribution of x and not to the distribution of the sample data. The distribution of the sample data is not supposed to approach a normal distribution, the DISTRIBUTION OF THE SAMPLE MEAN is what approaches a normal distribution

If np≥5 and nq≥​5, estimate P(more than 7) with n=12 and p=0.7 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.

The normal distribution cannot be used

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. The continuity correction is made to a discrete whole number x in the binomial distribution by representing the discrete whole number x by an interval.

A dataset includes a sample of 147 pulse rates of randomly selected women. Does that sample satisfy the following​ requirement: (1) The sample appears to be from a normally distributed​ population; or​ (2) the sample has a size of n​ > 30?

The requirement is satisfied because the sample size is greater than 30.

When women were finally allowed to become pilots of fighter​ jets, engineers needed to redesign the ejection seats because they had been originally designed for men only. The ejection seats were designed for men weighing between 140 lb and 211 lb. Weights of women are now normally distributed with a mean of 166 lb and a standard deviation of 45 lb. Complete parts​ (a) through​ (c) below.

a. If 1 woman is randomly​ selected, find the probability that her weight is between 140 lb and 211 lb. The probability is approximately 0.5603 b. If 27 different women are randomly​ selected, find the probability that their mean weight is between 140 lb and 211 lb. The probability is approximately 0.9987 c. When redesigning the ejection​ seat, which probability is more​ relevant? The part​ (a) probability is more relevant because the seat performance for a single pilot is more important.

The overhead reach distances of adult females are normally distributed with a mean of 205.5 cm and a standard deviation of 8.6 cm. a. Find the probability that an individual distance is greater than 218.00 cm. b. Find the probability that the mean for 15 randomly selected distances is greater than 203.70 cm. c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

a. The probability is 0.0735 b. The probability is 0.791 c. The normal distribution can be used because the original population has a normal distribution

In a study of 285,908 cell phone​ users, it was found that 69 developed cancer of the brain or nervous system. Assuming that cell phones have no​ effect, there is a 0.000283 probability of a person developing cancer of the brain or nervous system. We therefore expect about 81 cases of such cancer in a group of 285,908 people. Estimate the probability of 69 or fewer cases of such cancer in a group of 285,908 people. What do these results suggest about media reports that cell phones cause cancer of the brain or nervous​ system?

a. ​P(x≤69​)=0.1020 b. ​The media reports appear to be incorrect because one would expect that more than 81cell phone users would develop cancer. In​ fact, the study may offer some evidence to suggest that cell phone use decreases the probability of developing cancer. ANSWER FOR BE IS ALWAYS INCORRECT