BIOL361 Study Guide - Midterm Guide: Statistical Inference, Statistical Population, Interval Estimation
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Introduction: A Chi-square test is used to compare observed data with expected data according to a hypothesis. For instance, if you were crossbreeding 2 heterozygous pea plants, you would expect to see a 3:1 phenotypic ratio in the offspring. In this case, if you were to breed 400 pea plants, you would expect to see 300 plants showing the dominant trait and 100 showing the recessive trait. But what happens if you observe only 260 plants with the dominant trait and 140 plants with the recessive trait? Does this mean something is wrong with Mendelian genetics or is this difference in expected results just due to chance (random sampling error)? These are the questions that can be answered using Chi-square statistics. The results of this statistical test is used to either reject or accept (fail to reject) the null hypothesis. The null hypothesis states there is no significant difference between the observed results and the expected results. This means that if the null hypothesis is accepted, the difference in observed and expected results was just a matter of chance and so the observed results basically "fit" with what was expected. Degrees of freedom (df) = number of independent outcomes (Y) being compared less 1 df = Y-1 At the 95% confidence interval we are 95% confident that there is a significant difference between the observed and expected results, therefore rejecting the null hypothesis. Probability Value - Is the decimal value determined from the X2 table and is the probability of accepting the null hypothesis. A 0.05 probability value equates to a 95% confidence interval.
The Chi-squared test formula is: Example: If we cross two pea plants that are heterozygous yellow pods, we would expect a 3:1 phenotypic ratio. So let's say we actually did the cross and got 280 plants with green pods and 120 plants with yellow pods. Question: Is this a 3:1 phenotypic ratio? This is the value of Chi-squared Test. We have a total of 400 plants and we expect a 300 green:100 yellow phenotypic ratio If the calculated Chi-squared value is less than the critical value listed in the Chi-squared table, then we accept the null hypothesis. This means that there is no significant difference between the observed and the expected values. Our degrees of freedom (df) = 2 outcomes - 1, or df = 1. Now we go the X2 table below and using the df = 1 and probability value of 0.05, our critical value is 3.84. Since our calculated X2 value is 5.33, and is larger than the critical value, we reject the null hypothesis and can say (at 95% confidence) that there is a significant difference between our observed and expected values.
The parent generation is yellowed podded and green podded pea plants. You cross a yellow podded pea plant with a green podded pea plant and you get 100% yellow podded plants in the F1 Generation (Phenotypic ratio 4 : 0, yellow to green). What will be the expected phenotypic ratio when you allow the F1 generation to reproduce?
Fill out the Punnett square.
If we actually did the cross and got 1150 yellow and 350 green. Would this be a consistent with what was expected?
Learning Outcomes Questions
1. Why would you run a Chi-squared test?
| To determine if our data is consistent with expected results. | ||
| a To determine if our data is consistent with expected results. b To determine if our data exactly matches the expected results. | ||
| c To determine the expected results. | ||
| d | To compare the phenotypic ratios to the genotypic ratios. |
2. Determine the degrees of Freedom of the phenotypic ratio for this genetic cross.
a. 1
b. 2
c. 3
d. 4
e. 5
3. Using the data given, what is the result of your Chi-squared analysis? x2= ___.
| a. | 2.22 | |
| b | 2.71 | |
| c | 4.36 | |
| d | 187.78 | |
| e | 448.27 |
4. Using the results of your Chi-squared analysis, do we fail to reject or reject the null hypothesis?
| a. | Fail to reject the null | |
| b. | Reject the null | |
| c. | It cannot be determined from the data given |
This includes showing all calculations and explaining why you selected a specific answer for each multiple choice question (i.e. donât just circle the correct answer). I am looking to see how you got the answer, not just that you have the correct answer.
Below is data from the United States Cancer Statistics about the number of news cases of lung cancer in males in the U.S. for 2011 for ages 30 and older.
Table 1: U.S. Population Size and Incidence Cases of Lung Cancer in Males for 2011
| Age (Years) | U.S. Male Population | White Population | White Incidence Cases | Black Population | Black Incidence Cases |
| 30-34 | 10,181,564 | 7,950,924 | 88 | 1,383,650 | 22 |
| 35-39 | 9,666,162 | 7,557,263 | 212 | 1,271,897 | 23 |
| 40-44 | 10,363,126 | 8,259,215 | 643 | 1,324,847 | 132 |
| 45-49 | 10,856,131 | 8,766,225 | 2,032 | 1,373,346 | 444 |
| 50-54 | 10,976,172 | 9,004,383 | 5,128 | 1,329,551 | 1,148 |
| 55-59 | 9,733,849 | 8,101,629 | 8,607 | 1,091,030 | 1,687 |
| 60-64 | 8,466,308 | 7,183,960 | 12,340 | 845,362 | 2,067 |
| 65-69 | 6,014,777 | 5,178,255 | 15,792 | 539,796 | 2,077 |
| 70-74 | 4,360,942 | 3,769,569 | 16,011 | 376,129 | 1,864 |
| 75-79 | 3,204,196 | 2,807,216 | 14,319 | 255,221 | 1,426 |
| 80-84 | 2,319,839 | 2,078,071 | 11,284 | 154,155 | 883 |
| 85+ | 1,873,942 | 1,688,433 | 7,616 | 117,267 | 545 |
| Total | 88,017,008 | 72,345,143 | 94,072 | 10,062,251 | 12,318 |
3. What is the age-specific incidence rate for lung cancer in white men age 75-79 for 2011?(1 pt)
0.005 per 1000
1.3 per 1000
1.2 per 1000
5.1 per 1000
5.6 per 1000
F. Cannot be calculated from the given information
4. What is the age-specific incidence rate for lung cancer in black men age 75-79 for 2011? (1 pt)
0.005 per 1000
1.3 per 1000
1.2 per 1000
5.1 per 1000
5.6 per 1000
F. Cannot be calculated from the given information
Questions 5 through 7 are based on the information below:
Using the information in Table 1, researchers calculated the crude incidence rates and age-adjusted incidence rates (via the direct method) for both groups.
Table 2: Crude incidence rates and age-adjusted incidence rates for white and black males
| White Males | Black Males | ||
| Crude Incidence Rate | Age-adjusted Incidence Rate | Crude Incidence Rate | Age-adjusted Incidence Rate |
| 1.3 per 1000 | 1.2 per 1000 | 1.2 per 1000 | 1.5 per 1000 |
5. True or False, there is likely a difference in the age composition between the white and black males. Explain your answer
6. Based on the information in Table 2, it was reported that there was an increased risk of lung cancer in black males in 2011. (1 pt)
The conclusion is:
Correct, the age-adjusted rate is higher in black males compared to white males
Correct, because both the crude and age-adjusted rates are higher in black males compared to white males
Incorrect, because the incidence rate for white males is actually higher than the incidence rate for black males
7. What type of age-adjusted rate was calculated in table 2? How does it differ from the age-adjusted rate in question 5 in the in-class exercise for age-adjustments? Hint: See slide title: Adjustments for Other Measures of Disease Frequency in the lecture title Adjusted Rates (1 pt)
Question 8 is based on the information given below:
Cause-specific mortality rates for men in 2010 in the U.S. from various types of cancers are shown in the table below:
Cause-Specific Mortality Rates
(per 100,000 Men)
Location of the Cancer
in the Body White Asian
Pancreas 13.2 5.7
Prostate 19.1 2.5
Lung 62.3 18.5
Skin 4.9 0.3
8. The inference that White men in the U.S. are at higher risk of death due to cancer of the pancreas, prostate, lung, and skin compared to Asian men in the U.S. is
A. correct
B. incorrect because of failure to determine cause-specific mortality rates
incorrect because proportionate mortality alone does not give an estimate of risk
D. incorrect because of failure to adjust for differences in the age composition of the two populations
E. incorrect because counts were used when cause-specific mortality rates were needed
9. In January 1996, a team of epidemiologists identified a sample of 4,500 men, 65-74 years of age, for a study of prostate cancer. Tests indicated that 315 of the men already had prostate cancer and, therefore, were not at risk. The rest of the men were followed prospectively for five years to determine the incidence rate of prostate cancer in the sample. By the end of the five years of follow-up 156 of the men had developed prostate cancer. What is the incidence rate of prostate cancer in this group?
10. An epidemiologic investigation that started on January 1, 2011, identified a population of 1,000 individuals. At the start of the study 4 were found to have the disease. During the year of study, 6 new cases were found; thus a total of 10 cases were identified. Among those 10 cases, 6 deaths occurred during the year. What was the prevalence of the disease during the 1-year study?
