please answer the following questions:
Ch 3 The Normal Distributions 1. Which of the following is not true for a Normal density curve? a. It is symmetric so the median is the same as the mean. b. It is shaped like a bell. C. The mean of the distribution can be negative, zero, or positive. d. The standard deviation must be 1. 2. Larger values of the standard deviation result in a Normal curve that is Hint: See Fig 3.8 on p81. a. shifted to the right b. shifted to the left C. narrower and more peaked d. wider and flatter 3. Which of the following statements is correct? Hint: Read the following in your book: the box Normal Distributions on p81 the 2nd paragraph "If the variable...." and the box Standard Normal Distribution on p86 the last two sentences "However, many sets of ...." in the Ist paragraph "Why are the Normal distributions...." on p82 I. Normal distributions are a type of continuous probability distribution. II. There is more than one Normal distribution and each Normal distribution is completely specified (or identified) by two numbers: its mean / and standard deviation o. III. A standard Normal distribution is a special type of Normal distributions i.e. it is a Normal distribution with # = 0 and o = 1. IV. Standardizing a Normal variable X produces a new random variable, denoted by z, that has the standard Normal distribution. (That is, if X is Normally distributed, then we can convert an observation x to a z-value by the formula z = = and z is Normally distributed with # = 0 and ( = 1.) V. The z-scores (or z-values) of the standard Normal distribution ranges from positive infinity to negative infinity. - 2 -VI. A researcher should not assume that the data collected are automatically Normally distributed- the researcher should first plot the data to determine if the histogram, stemplot or dotplot is roughly bell-shaped before using the Normal distribution to calculate probabilities. a. I, II, V, VI b. II, III, IV, V, VI c. I, II, III, IV d. I, II, III, IV, V, VI 4. Which of the following statements is true about continuous probability distributions? Hint: Continuous probability distributions are represented by density curves, which are based on histograms. If the histogram is symmetric, the mean = median. However, not all histograms or density curves are symmetric. See the diagrams on p79. a. The shape of the density curve must be symmetric. b. The total area under the density curve must be exactly 1. c. The mean and the median are the same for the density curve. d. All of the above. 5. For a density curve with a long left tail, which of the following is true for the mean and median? Hint: Recall that density curves are based on histograms. For this question, imagine that you have a histogram that has a long left tail. See the diagrams on p79. a. mean = median b. mean > median c. mean 140) is as follows: First, let's assume that before collecting data, we, as researchers, have decided that a probability smaller than 1% is "a small probability". That is, we consider an event with a probability smaller than 1% very unlikely to happen out of pure chance alone. (Now, of course, other researchers might choose a different percentage as "small" e.g. 2%, 6%, etc.) Assume that the population mean is 125 mg/dl. If we imagine that we repeat taking all possible samples of size 4 from the population, then only about 1.3 out of sample means will have values as large as _cm or even larger. In other words, if the population mean is indeed 125 mg/dl, the chances that such extremely large sample means would occur by pure chance - 12 - alone are small (since the probability calculated from the question above is smaller than our threshold of 1%). To put it another way, if the population mean is indeed 125 mg/dl, then we would NOT expect a sample mean as large as to occur if we only select one single sample. (But if such an extreme sample mean does occur, it implies that there is something else, other than pure chance, is at play. That "something else" is that the assumption could be wrong after all because if the assumption is true, such an extreme sample mean should not happen. This "rare event" interpretation forms the reasoning behind hypothesis testing, an important topic for the second half of the course.) Hint: See Lab work 4/8. 0.1 = - (1 out of 10) 0.01 = = (1 out of 100), etc. a. 10, 140, 140 c. 1000, 140, 140 b. 100, 140, 140 d. 1000, 125, 140 32. Left-tail probability of X A large school district in southern California asked all its eighth graders to measure the length of their right foot at the beginning of the school year, as part of a science project. The data show that foot length is approximately Normally distributed with mean of 23.4 cm and a standard deviation of 1.7 cm. Suppose that 25 eighth graders from this population are randomly selected. What is the probability that the sample mean foot length is less than 22.62 cm? Hint: Find P (x 140). So you need to use the z formula in Ch 3. a. 0.3173 c. 0.1587 b. 0.2525 d. 0.0668 (This question is a modified version of #15.28 a) on p372 in our textbook.)
