Investigation 1.17: Cat Households

Section 4.8 Investigation 1.17: Cat Households

Introduction image for Cat Households investigation

The American Veterinary Medical Association (AVMA) has conducted a Pet Demographics Survey about every five years since the early 1980s. In spring 2024, a total of 7,539 people completed an opt-in, online survey. After "results have been weighted based on certain demographic and other variables to match their distribution in the U.S. population and household information reported by the U.S. Census Bureau," the AVMA reported that 32.1% of households owned at least one cat.

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Checkpoint 4.8.1. Identify observational units and variable.

Identify the observational units and variable of interest.

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Observational units:

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Variable:

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Solution.

Observational units: American households

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Variable: Whether or not there is a cat in the household

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Checkpoint 4.8.2. Identify parameter or statistic.

Is 32.1% a parameter or a statistic?

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statistic
parameter

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Checkpoint 4.8.3. Identify symbol.

Indicate the symbol used to represent 0.321.

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\(\hat{p}\)
\(\pi\)
\(\hat{\rho}\)
\(\mu\)
\(\bar{x}\)

Hint.

Think about whether this is a sample statistic or population parameter.

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Solution.

Statistic, \(\hat{p}\text{,}\) because it is computed from the sample

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Checkpoint 4.8.4. Conduct hypothesis test.

Conduct a test of whether the sample data provide evidence that the population proportion who own a pet cat differs from 1/3. State the hypotheses, and report the standardized statistic and p-value. State your test decision at the \(\alpha = 0.05\) level, and summarize your conclusion in the context of this study.

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Solution.

Let \(\pi\) represent the proportion of all American households with a cat (in spring 2024)

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\(H_0: \pi = 1/3\) (population proportion equals one-third)

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\(H_a: \pi \neq 1/3\) (differs from one-third)

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Because the sample size is large and because the population size is large, I will use a z-test.

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\(\hat{p} = 0.321\text{,}\) \(n = 7539\)

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Standard error: \(SE = \sqrt{\frac{(1/3)(2/3)}{7539}} \approx 0.00543\)

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Test statistic: \(z = \frac{0.321 - 0.3333}{0.00543} \approx -2.27\)

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P-value: \(\approx 0.0234\) (two-sided)

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With a large standardized statistic and a small p-value, we will reject the null hypothesis and conclude that we have convincing evidence at the 0.05 level (0.0234 < 0.05) that the proportion of American households (in spring 2024) with a cat differed from 1/3.

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Statistical software output for hypothesis test — Figure 4.8.5. Hypothesis test output
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Checkpoint 4.8.6. Explain small p-value.

Explain why the p-value turns out to be so small, when the sample proportion of households with a cat seems to be quite close to one-third.

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Solution.

The p-value is so small because the sample size is so large.

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Checkpoint 4.8.7. Calculate confidence interval.

Produce a 95% confidence interval (CI) for the population proportion who own a pet cat. Interpret this interval.

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Solution.

Using \(\hat{p} = 0.321\) and \(n = 7539\text{:}\)

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\(SE(\hat{p}) = \sqrt{\frac{0.321(0.679)}{7539}} \approx 0.00538\)

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95% CI: \(0.321 \pm 1.96(0.00538) = 0.321 \pm 0.0105 = (0.3105, 0.3315)\)

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We are 95% confident that between 31.05% and 33.15% of American households owned a pet cat in 2024.

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Statistical software output for confidence interval — Figure 4.8.8. Confidence interval output
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Checkpoint 4.8.9. Compare CI to test decision.

Is the confidence interval consistent with the test decision about 1/3? Explain.

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Solution.

This is consistent because the confidence interval does not include 0.3333.

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Checkpoint 4.8.10. Strong evidence not one-third?

Do the sample data provide very strong evidence that the population proportion who own a pet cat is not one-third? Explain whether the p-value or the CI helps you to decide.

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Solution.

Yes, the evidence is very strong because the p-value is small.

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Checkpoint 4.8.11. Very different from one-third?

Do the sample data provide strong evidence that the population proportion who own a pet cat is very different from one-third? Explain whether the p-value or the CI helps you to decide.

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Solution.

No, the confidence interval is very close to 1/3, so even though we have strong evidence \(\pi\) differs from one-third, the confidence interval shows us that it’s not different by a lot, that is, not in a practical sense.

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Discussion.

Keep in mind the difference between statistical significance and practical significance. With large sample sizes, sample proportions will vary little from sample to sample, and so even small differences (that may seem minor to most of us) will be statistically significant. Saying that a sample result is unlikely to happen by chance (and therefore is statistically significant) is not the same as saying the result is important or even noteworthy (practically significant), depending on the context involved.

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You have attempted of activities on this page.

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