Stat200 Introduction To Statistics Project 5 Given The Data

Stat200 Introduction To Statistics Project 5given The Data And Informa

Given the data and information in the following examples, determine the appropriate hypothesis test, state the null hypothesis, and identify the random sample. The possible tests are One Sample Z-test, One Sample Proportion Test, or One Sample T-test. You are not required to perform the tests, only to identify the correct test, null hypothesis, and describe the sample for each scenario.

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1. In 1999, the average percentage of women who received prenatal care per country was 80.1%. A sample of countries in 2009 shows various percentages of women receiving prenatal care. Determine whether the data suggest an increase in the average percentage of women receiving prenatal care in 2009 compared to 1999 at the 5% significance level. Identify the appropriate test, formulate the null hypothesis, and specify the random sample.

2. Eyeglassomatic manufactures lenses and finds that 11% of all lenses are defective. In a three-month period, out of 34,641 defective lenses, 5,865 were due to scratches. Determine if there are significantly more defects from scratches than from other causes at the 1% significance level. State the appropriate test, the null hypothesis, and describe the sample.

3. In the United States, males aged 40-49 typically consume an average of 103.1 grams of fat daily with a standard deviation of 4.32 grams. A sample of 30 males attending a football game is observed. Assess whether these males consume more fat on average than the general population at the 5% significance level by determining the appropriate test, null hypothesis, and the nature of the sample.

4. A study of Atlantic cod caught in Karlskrona found the mean length to be 49.9 cm with a standard deviation of 3.74 cm. A random sample of 30 fish from the "Gale Force" fleet is examined. Evaluate if this fleet catches larger fish than the fleet average at the 5% significance level, specifying the test, null hypothesis, and sample description.

5. According to a 2008 FTC report, 23% of all complaints in 2007 involved identity theft. Arkansas had 1,601 complaints of identity theft out of 3,482 consumer complaints. Determine if Arkansas's proportion of identity theft complaints is higher than 23% at the 5% significance level. Identify the appropriate test, null hypothesis, and sample details.

6. Pulse rates after running are collected from females who drink alcohol. The mean pulse rate for females who do not drink is 97 beats per minute. Analyze whether females who drink have a higher mean pulse rate at the 5% significance level by selecting the test, formulating the null hypothesis, and describing the sample.

7. The average starting salary for nurses nationally is $67,694 with a standard deviation of roughly $10,333. A sample of 42 nurses from Tampa Bay hospitals is analyzed to see if their average salary is lower than the national average at the 5% level. Define the appropriate test, null hypothesis, and the sample.

8. A Gallup poll showed 81% of Americans believed there was a conspiracy behind Kennedy's assassination in 2001. In 2013, among 1,039 adults surveyed, 634 believed in a conspiracy. Determine if the proportion of believers decreased at the 1% significance level, and specify the test, null hypothesis, and sample.

9. In Florida lakes, the mercury content in fish was tested across 53 lakes, with the sample mean being above the FDA limit of 1.0 mg/kg. Evaluate whether the mercury levels are significantly higher than the allowable limit at the 10% significance level, identifying the test, null hypothesis, and data collection details.

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The process of statistical hypothesis testing involves selecting an appropriate test based on the nature of the data and research question, formulating a null hypothesis that presumes no effect or no difference, and using sample data to determine whether to reject this hypothesis at a specified significance level. Each of the scenarios presented requires a tailored approach considering the data type, sample size, and assumptions about the distribution, like normality or approximate normality.

Question 1: Prenatal Care Percentage Increase

Given the data on percentages of women receiving prenatal care in 2009, compared with the 1999 national average of 80.1%, the scenario involves assessing whether the recent data indicates an increase. Since the data comprises numerical percentages and the sample is taken from multiple countries, a one-sample t-test is appropriate because the population standard deviation is unknown, and the sample size may be limited or not guaranteed to be normally distributed. The null hypothesis states that the mean percentage in 2009 is less than or equal to 80.1%, indicating no increase. The sample is randomly selected from the set of countries included for 2009 data.

Question 2: Defects in Lens Manufacturing

The objective is to test for an excess of scratches among defective lenses, based on a large sample size (34,641) and the known proportion of scratches (11%). Since we are comparing sample proportion to a known population proportion, a one-sample proportion z-test is appropriate. The null hypothesis posits that the proportion of scratches in the sample equals the population proportion of 11%. The sample consists of defective lenses identified over a predetermined time frame, with the number of scratches recorded.

Question 3: Fat Consumption in Males Attending Football Game

The question involves determining whether the average fat intake of males attending a football game exceeds the typical population mean of 103.1 g. Here, because the population standard deviation is unknown, but the sample size is 30 and the data is approximately mound-shaped, a one-sample t-test is suitable. The null hypothesis claims that the mean fat consumption for this group is less than or equal to 103.1 g. The sample is a simple random sample of males attending a football game.

Question 4: Fish Lengths Caught by a Fleet

Assessing whether a specific fleet catches larger fish involves comparing the sample mean fish length to the known population mean. Since the population standard deviation is known, and the fish lengths are approximately normally distributed, a one-sample z-test is appropriate. Null hypothesis states the mean length of fish caught by the fleet is less than or equal to the population mean of 49.9 cm. The sample includes 30 fish caught by the Gale Force fleet.

Question 5: Proportion of Identity Theft Complaints in Arkansas

Evaluating whether Arkansas has a higher proportion of identity theft complaints than the overall proportion involves a comparison of sample proportion to a known proportion. Because the data involves proportions, a one-sample z-test for proportions is suitable. The null hypothesis asserts that Arkansas's proportion does not differ from 23%. The sample comprises the reported complaints from Arkansas.

Question 6: Pulse Rates of Women Who Drink Alcohol

To test whether women who consume alcohol have a higher mean pulse rate than women who do not, a one-sample t-test is appropriate, assuming the data is approximately normally distributed. The null hypothesis states that the mean pulse rate of women who drink is less than or equal to 97 beats per minute. The sample is randomly selected from females who drink alcohol.

Question 7: Nurses’ Starting Salaries in Tampa Bay

Testing whether Tampa Bay nurses earn a lower starting salary than the national average involves a one-sample z-test, given the known population standard deviation and sample data. Null hypothesis presumes that the mean salary in Tampa Bay is at least equal to the national mean. The sample consists of 42 randomly selected nurses.

Question 8: Belief in Kennedy Conspiracy

To examine if the proportion of Americans believing in a conspiracy has decreased from 2001 to 2013, a two-proportion z-test could be performed. However, since only data from 2013 is available with a comparison to the previous proportion, a one-proportion z-test is appropriate, testing whether the current proportion is less than 81%. Null hypothesis states that the proportion remains the same or higher. The sample is the 1,039 surveyed adults in 2013.

Question 9: Mercury Levels in Florida Lakes

Evaluating whether the mean mercury content exceeds the FDA limit involves a one-sample t-test, considering the sample mean, standard deviation, and approximate normality. Null hypothesis states that the mean mercury level is less than or equal to 1.0 mg/kg. The data includes measurements from 53 lakes.

In summary, the scenarios primarily involve selection between one-sample z-tests or t-tests based on whether population standard deviations are known and distribution assumptions, as well as proportion tests for categorical data. Proper formulation of hypotheses allows for systematic assessment of each situation to reach valid statistical conclusions regarding the populations involved.

References

  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
  • DeGroot, M. H., & Schervish, M. J. (2012). Probability and Statistics (4th ed.). Pearson.
  • Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses (3rd ed.). Springer.
  • Moore, D. S., Notz, W. I., & Flinger, M. A. (2013). Statistics: Concepts and Controversies (8th ed.). W. H. Freeman & Co.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics (8th ed.). Pearson.
  • Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis (6th ed.). Brooks/Cole.
  • Salman, A., & Shaikh, M. (2014). Understanding hypothesis testing: A tutorial approach. Journal of Applied Statistics, 41(2), 291-312.
  • Vandendriessche, B., & Verdonck, M. (2018). Applied Statistics: From Data to Conclusions. Springer.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Zar, J. H. (2010). Biostatistical Analysis (5th ed.). Pearson.

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