Normal Probability Distributions

Which distributions have the same mean? Check all that apply.

• red

• yellow

• green

• blue

• none of them have the same mean

Clara recorded 50 numerical observations on a certain variable and then calculated the mean <x> and the standard deviation "s" for the observations. To help decide whether a normal model is appropriate, she created the chart. In Clara’s chart, the letters a,b,c,d,and e represent the number of observations falling in each interval. Which of the following list of counts for a,b,c,d,and e, respectively, is the best indicator that the variable can be modeled with a normal approximation

• 1, 7, 34, 7, 1

• 1, 10, 28, 10, 1

• 2, 8, 30, 8, 2

• 2, 4, 38, 4, 2

• 5, 5, 30, 5, 5

Carly commutes to work, and her commute time is dependent on the weather. When the weather is good, the distribution of her commute times is approximately normal with mean 20 minutes and standard deviation 2 minutes. When the weather is not good, the distribution of her commute times is approximately normal with mean 30 minutes and standard deviation 4 minutes. Suppose the probability that the weather will be good tomorrow is 0.9. Which of the following is closest to the probability that Carly’s commute time tomorrow will be greater than 25 minutes?

• 0.0056

• 0.0894

• 0.0950

• 0.8055

• 0.9006

A machine is used to fill bags with a popular brand of trail mix. The machine is calibrated so the distribution of the weights of the bags of trail mix is normal, with mean 240 grams and standard deviation 3 grams. Of the following, which is the least weight of a bag in the top 5 percent of the distribution?

• 244 grams

• 242 grams

• 243 grams

• 245 grams

• 248 grams

A manufacturer of cell phone batteries claims that the average number of recharge cycles for its batteries is 400. A consumer group will obtain a random sample of 100 of the manufacturer’s batteries and will calculate the mean number of recharge cycles. Which of the following statements is justified by the central limit theorem?

• The distribution of the number of recharge cycles for the sample is approximately normal because the population mean of 400 is greater than 30.

• The distribution of the number of recharge cycles for the sample is approximately normal because the sample size of 100 is greater than 30.

• The distribution of the number of recharge cycles for the population is approximately normal because the sample size of 100 is greater than 30.

• The distribution of the sample means of the number of recharge cycles is approximately normal because the sample size of 100 is greater than 30

• The distribution of the sample means of the number of recharge cycles is approximately normal because the population mean of 400 is greater than 30.

The mean and standard deviation of the sample data collected on continuous variable x are −0.25 and 0.03, respectively. The table shows the relative frequencies of the data in the given intervals. Based on the table, do the data support the use of a normal model to approximate population characteristics?

• Yes, because the sum of the relative frequencies is 1.00.

• Yes, because the distribution of relative frequencies is very close to the empirical rule for normal models.

• No, because the values are negative and normal models are used for positive values.

• No, because the distribution of relative frequencies is very far from the empirical rule for normal models.

• No, because the sample size and the population parameters are not known.

The continuous random variable N has a normal distribution with mean 7.5 and standard deviation 2.5. For which of the following is the probability equal to 0 ?

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Data will be collected on the following variables. Which variable is most likely to be approximated by a normal model?

• The distribution of the number of books read last week by middle school students, where the right tail of the distribution is longer than the left

• The distribution of life span, in minutes, for batteries of a certain size, where most life spans cluster around the center of the distribution but with some very low and some very high life spans

• The distribution of ages, in years, of the students at a certain college, where most students are between 18 and 22 years old, but ages greater than 22 will probably be more spread out than ages less than 18

• The distribution of the number of birthdays per month for the employees at a certain company, where the number of birthdays in each month is approximately equal

• The distribution of the length of a stay, in days, in a hospital after surgery, where many patients have very short hospital stays, but some stays are quite lengthy and considered high outliers

According to government data, 22 percent of children in the United States under the age of 6 years live in households with incomes that are classified at a particular income level. A simple random sample of 300 children in the United States under the age of 6 years was selected for a study of learning in early childhood. If the government data are correct, which of the following best approximates the probability that at least 27 percent of the children in the sample live in households that are classified at the particular income level? (Note: z represents a standard normal random variable.)

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A recent study was conducted to investigate the duration of time required to complete a certain manual dexterity task. The reported mean was 10.2 seconds with a standard deviation of 16.0 seconds. Suppose the reported values are the true mean and standard deviation for the population of subjects in the study. If a random sample of 144 subjects is selected from the population, what is the approximate probability that the mean of the sample will be more than 11.0 seconds?

• 0.1151

• 0.2743

• 0.7257

• 0.8849

• Based on the values of the true mean and true standard deviation, it can be concluded that the population distribution is not normal and therefore the probability cannot be calculated.