Lecture 6 – Z scores



Lecture 6 – Z scores

An Introduction to Making Inferences

descriptive statistics – summarize important characteristics of known population data

inferential statistics – we use sample data to make inferences or generalizations about a population (review population and sample)

sampling statistic - any statistic that describes the distribution of values for a variable, or relationships between variables in a sample

population parameter – the estimated characteristics of a population derived from sampling statistics

➢ must have a well chosen sample!!! (sound sampling procedures)

z-scores (and the normal distribution) enable us to standardize values so they can be compared (example: SAT); also remember 68-95-99.7 rule

looking at data: the smaller the standard deviation in relation to the range of responses, the more homogeneous are the responses

standard score (or z-score) – the number of standard deviations that a given value x is above or below the mean. In a z distribution, the mean = 0, and the standard deviation is 1. Thus, a z-score of 1.5 is 1 ½ sd above the mean, and a z-score of –2 is 2 sd below the mean.

sample z: [pic] population z: [pic]

Examples:

Suppose you took 3 midterm examinations in Math, English, and Biology, and you want to compare your scores:

| |Math |English |Biology |

|your score |90 |85 |93 |

|Mean |85 |82 |94 |

|Standard deviation |5 |2 |1 |

|z-score | | | |

1. Find the following probabilities using Appendix A page 469-473:

z < 1.56 z < -.68 z > 2.34

z between 0 and 2.1 z between –1.23 and .90

2. What proportion of all young women are less than 68 inches tall? This proportion is the area under the normal curve with a mean of 64.5 and standard deviation of 2.5”.

3. The level of cholesterol in the blood is important because high cholesterol levels may increase the risk of heart disease. The distribution of blood cholesterol levels in a large population of people of the same age and sex is roughly normal. For 14-year old boys, the mean is 170 mg of cholesterol per deciliter of blood (mg/dl) and the standard deviation is 30mg/dl. Levels above 240 mg/dl may require medical attention. What percent of 14 year-old boys have more than 240 mg/dl of cholesterol?

What percent of 14-year-old boys have blood cholesterol between 140 and 200 mg/dl?

4. Scores on the SAT verbal test in recent years follow approximately the normal distribution, with mean of 505 and sd 110. How high must a student score in order to place in the top 10% of all students taking the SAT?

5. One classic use of the normal distribution is inspired by a letter to “Dear

Abby” in which a wife claimed to have given birth 308 days after a brief visit from her husband, who was serving in the Navy. The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. Given this information, find the probability of a pregnancy lasting 308 days or longer. What does the result suggest?

6. Domino sugar packets are labeled as containing 3.5 g. Assume that those packets are actually filled with amounts that are normally distributed with a mean of 3.586 g and a standard deviation of 0.074 g. What percentage of packets have less than 3.5 g? Are many consumers being cheated?

7. A study compared the facial behavior of non-paranoid schizophrenic persons with that of a control group of normal persons. The control group was timed for eye contact during a period of 5 minutes, or 300 seconds. The eye-contact times were normally distributed with a mean of 184 sec and a standard deviation of 55 sec (based on data from “Ethological Study of Facial Behavior in Non-paranoid and Paranoid Schizophrenic Patients,” by Pitman, Kolb, Orr, and Singh, Psychiatry, 114:1). Because results showed that non-paranoid schizophrenic patients had much lower eye-contact times than did the control group, you have decided to further analyze people in the control group who are in the bottom 5%. For the control group, find the eye-contact time separating the bottom 5% from the rest.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download