In A Survey, 32 Of The Respondents Stated They Talk To The
1in A Survey 32of The Respondents Stated That They Talk To Their
In a survey, 32% of respondents reported talking to their pets on the answering machine or telephone. A veterinarian found this result suspiciously high and conducted a random sample of 250 pet owners, discovering that 70 of them spoke to their pets on the answering machine or telephone. The question is whether the veterinarian should be skeptical, based on the data, using a significance level of α = 0.05. The hypothesis framework is:
Null hypothesis (H₀): p = 0.45 (the proportion of pet owners talking to pets is 45%)
Alternative hypothesis (H₁): p
The sample size (n) is 250, and the number of pet owners who talk to their pets (x) is 70.
To evaluate this, we will perform a hypothesis test for proportions, both through the classical (critical value) approach and the P-value approach, at α = 0.05.
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To assess whether the proportion of pet owners who talk to their pets on the answering machine genuinely exceeds a given benchmark, the veterinarian’s skepticism prompts us to examine the data through hypothesis testing. Specifically, the hypotheses are set as H₀: p = 0.45 versus H₁: p
The data collected consists of a sample of 250 pet owners, among whom 70 reported talking to their pets on the answering machine or telephone. This yields a sample proportion (p̂) of 70/250 = 0.28. The observed sample proportion is notably lower than the hypothesized 0.45, suggesting that the initial survey's figure might be overestimated if the true proportion is less than 45%. To verify whether this discrepancy is statistically significant, we conduct a hypothesis test for a population proportion.
Using the classical approach, the first step involves calculating the standard error (SE) of the sampling distribution under the null hypothesis:
SE = √[p₀(1 - p₀) / n] = √[0.45 * 0.55 / 250] ≈ 0.0314.
The test statistic (z) is calculated as:
z = (p̂ - p₀) / SE = (0.28 - 0.45) / 0.0314 ≈ -5.42.
Since the alternative hypothesis is that p
Because the calculated z = -5.42 is far less than -1.645, we reject the null hypothesis, indicating that the proportion of pet owners who talk to their pets is statistically significantly less than 45%. This contradicts the veterinarian’s suspicion of a high rate, further supporting skepticism about the initial survey's figure.
For the P-value approach, the P-value corresponding to z = -5.42 is essentially zero (p
In conclusion, both testing methods strongly suggest that the true proportion of pet owners talking to their pets on answering machines or telephones is significantly less than 45%. Therefore, the veterinarian’s skepticism regarding the high survey percentage is statistically justified.
Additional consideration involves the assumptions of the hypothesis test: the sample size is sufficiently large for normal approximation (np₀ and n(1 - p₀) both greater than 5), which they are in this case. Moreover, since the data were randomly sampled, the inference is valid.
Overall, the evidence indicates the initial reported rate was indeed overstated, and the skepticism was warranted.
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