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  • Investigation 1.8: Halloween Treat Choices

  • Obesity has become a widespread health concern, especially in children. Researchers believe that giving children easy access to food increases their likelihood of consuming extra calories. Schwartz, Chen, and Brownell (2003) examined whether children would be willing to take a small toy instead of candy at when trick-or-treating on Halloween. They had seven homes in 5 different towns in Connecticut present children with a plate of 4 toys (stretch pumpkin men, large glow-in-the-dark insects, Halloween theme stickers, and Halloween theme pencils) and a plate of 4 different name brand candies (lollipops, fruit-flavored chewy candies, fruit-flavored crunchy wafers, and “sweet and tart” hard candies) to see whether children were more likely to choose the candy or the toy. The houses alternated whether the toys were on the left or on the right. Data were recorded for 284 children between the ages of 3 and 14 (who did not ask for both types of treats). 

     

  • (a) Identify the observational unit: Variable:    

  • (c) State an appropriate null and alternative hypothesis involving this parameter (in symbols or in words), for testing whether there is strong evidence of a preference for either the toys or the candy.
    Null hypothesis Ho:
    Alternative hypothesis Ha:

  • (d) What does “theory” predict for the mean and standard deviation of the distribution of “number of successes” under the null hypothesis if we model this experiment as a binomial process?
    Mean
    Standard deviation

  • Normal approximation to Binomial distribution

  • The normal distribution has many, many applications to quantitative variables in general (e.g., many biological variables like height are assumed to follow a normal distribution). For now, we will focus on using this mathematical model as an approximation to the null distribution.

    Central Limit Theorem: If the sample size is large enough, then the distribution of “number of successes” for a binomial random variable will be well-modeled by the normal probability distribution. The sample size is considered large enough if 𝑛×𝜋≥10 and 𝑛×(1−𝜋)≥10. 

     

  • (g) Are these criteria met for this study? (Assume the hypothesized value for 𝜋)
    Yes or no? , because n x 𝜋 = which is than 10 and n x (1 - 𝜋) = which is than 10.

  • (h) In the applet, change the Choose Statistic radio button from Number of heads to Proportion of heads. Report the new values for the mean and standard deviation of the null distribution for this statistic and report the new p-value.
    Mean:
    Standard deviation:
    p-value:

  • Use the One Proportion Inference applet to

    • display the exact binomial distribution for n = 284 and 𝜋 = 0.50
    • verify the values for the mean and standard deviation
    • determine the p-value for your hypotheses in (c) (Recall 135 children chose the toy and 149 chose the candy)
    • confirm that the conclusion you would draw from this p-value matches the conclusion you drew from the standardized statistic

     

     

  • One Proportion z-test

  • Definition: The One Proportion z-test calculates the standardized statistic as

     where  𝜋0  is the hypothesized probability

    and uses the standard normal distribution (mean 0, std dev 1) to find the p-value. 

    The One Proportion z-test is more well-known than the Exact Binomial test because historically it was much easier to find the probability from the standard normal distributon (e.g., a table) than to calculate and sum a bunch of binomial probabilities.  If you decide you want to use the normal approximation (e.g., you have a larger sample size), then you can go straight to the Theory-Based Inference applet:

     

     

    • Enter the sample information and press the Calculate button.
    • Check the Test of significance box.
    • Specify the form of the alternative hypothesis.
    • Confirm your earlier results (within a bit of rounding discrepency).

    While this normal approximation should work, we can actually do better.  We can use a "continuity correction" to help the probability from the normal distribution look more like the probability from the binomial distribution.

    • Check the continuity correction box.
    • (Repeatedly uncheck and recheck the box to see what's happening)

  • Conclusions

  • Write a paragraph summarizing

    • the sample measured, the observed statistic, and the sample size
    • the test of significance (the conclusion about 0.50 and your evidence, i.e., cite your p-value, "We do/do not have evidence that...")
    • the confidence interval (interpret the interval in context, "We are 95% confident that ... ")
    • also indicate to which "process"/population you are generalizing your conclusions.
  • Should be Empty: