Vocabulary and Notation 341 32. Transformation to normality: Consider the following data set: 28.0 6.7 8.6 2.3 25.0 12.5 4.4 37.3 12.0 48.0 0.7 11.6 0.1 a. Is it reasonable to treat it as a sample from an approximately normal population? No b. Perform a square-root transformation. Is it reasonable to treat the square-root-transformed data as a sample from an approximately normal population? Yes c. Perform a reciprocal transformation. Is it reasonable to treat the reciprocal-transformed data as a sample from an approximately normal population? No Answers to Check Your Understanding Exercises for Section 7.6 1. a. The plot contains an outlier. The population is not approximately normal. b. We may treat this population as approximately normal. 2. a. We may treat this population as approximately normal. b. The histogram has more than one mode. The population is not approximately normal. 3. The plot reveals that the sample is strongly skewed. The population is not approximately normal. 4. There are no outliers and no evidence of strong skewness. We may treat this population as approximately normal. 5. The points follow a straight line fairly closely. We may treat this population as approximately normal. 6. The points do not follow a straight line fairly closely. The population is not approximately normal. Chapter 7 Summary Section 7.1: Continuous random variables can be described with probability density curves. The area under a probability density curve over an interval can be interpreted in either of two ways. It represents the proportion of the population that is contained in the interval, and it also represents the probability that a randomly chosen value from the population falls within the interval. The normal curve is the most commonly used probability density curve. The standard normal curve represents a normal population with mean 0 and standard deviation 1. We can find areas under the standard normal curve by using Table A.2 or with technology. Section 7.2: In practice, we need to work with normal distributions with different values for the mean and standard deviation. Technology can be used to compute probabilities for any normal distribution. We can also use Table A.2 to find probabilities for any normal distribution by standardization. Standardization involves computing the z-score by subtracting the mean and dividing by the standard deviation. The z-score has a standard normal distribution, so we can find probabilities by using Table A.2. Section 7.3: The sampling distribution of a statistic such as a sample mean is the probability distribution of all possible values of the statistic. The Central Limit Theorem states that the sampling distribution of a sample mean is approximately normal so long as the sample size is large enough. Therefore, we can use the normal curve to compute approximate probabilities regarding the sample mean whenever the sample size is sufficiently large. For most populations, samples of size 30 or more are large enough. Section 7.4: The Central Limit Theorem can also be used to compute approximate probabilities regarding sample proportions. The sampling distribution of a sample proportion is approximately normal so long as the sample size is large enough. The sample size is large enough if both np and n(1 − p) are at least 10. Section 7.5: A binomial random variable represents the number of successes in a series of independent trials. The number of successes is closely related to the sample proportion, because the sample proportion is found by dividing the number of successes by the number of trials. Since the sample proportion is approximately normally distributed whenever np ≥ 10 and n(1 − p) ≥ 10, the number of successes is also approximately normally distributed under these conditions. Therefore, the normal curve can also be used to compute approximate probabilities for the binomial distribution. Because the binomial distribution is discrete, the continuity correction can be used to provide more accurate approximations. Section 7.6: Many statistical procedures require the assumption that a sample is drawn from a population that is approximately normal. Although it is very difficult to determine whether a small sample comes from such a population, we can examine the sample for outliers, multimodality, and large degrees of skewness. If a sample contains no outliers, is not strongly skewed, and has only one distinct mode, we will treat it as though it came from an approximately normal population. Dotplots, boxplots, histograms, stem-and-leaf plots, and normal quantile plots can be used to assess normality. Vocabulary and Notation Central Limit Theorem 313 normal quantile plot 336 standard normal curve 289 continuity correction 326 population proportion 318 standard normal distribution 289 normal approximation to binomial 326 probability density curve 286 standardization 299 normal curve 288 sample proportion 318 z-score 299 normal distribution 288 standard error 311
navidi_monk_elementary_statistics_2e_ch7-9
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