Question 1 In A Survey Of 380 Households Regarding The Owner ✓ Solved

Question 1in A Survey Of 380 Households Regarding The Ownership Of Vcr

In a survey of 380 households regarding the ownership of VCRs and DVD players, the following data was obtained: 350 households own one or more VCRs. 180 households own one or more VCRs and one or more DVD players. 13 households do not own a VCR or a DVD player. How many households only own one or more DVD players?

Sample Paper For Above instruction

The survey conducted among 380 households provides valuable insights into the ownership patterns of VCRs and DVD players. To analyze the data effectively, we employ set notation and principles of basic set theory to determine the number of households that exclusively own DVD players. The data indicates that 350 households own at least one VCR, while 180 households own both VCRs and DVD players. Additionally, 13 households do not own either device, which means 367 households own at least one of these devices (since 380 total households minus 13 who own neither).

Let’s define the following sets:

  • V = the set of households owning VCRs.
  • D = the set of households owning DVD players.

From the data, we know:

  • |V| = 350
  • |V ∩ D| = 180
  • Households owning neither = 13

We are asked to find the number of households that only own DVD players, which can be represented as |D \ V|. To do this, first determine the total number of households owning DVD players, |D|. Since |V ∩ D| = 180 and total households that own at least one device is (380 - 13) = 367, we can find |D| using the inclusion-exclusion principle.

The total number of households owning at least one device is:

|V ∪ D| = |V| + |D| - |V ∩ D|

And we know that:

|V ∪ D| = 367 (since 13 households own neither)

Rearranged, this becomes:

367 = 350 + |D| - 180

Solving for |D| gives:

|D| = 367 - 350 + 180 = 197

Now, the number of households that only own DVD players is:

|D only| = |D| - |V ∩ D| = 197 - 180 = 17

Therefore, 17 households own only DVD players. This matches option (b) 17 households.

In conclusion, the survey reveals that a significant portion of households own both devices, but a small group owns only DVD players, highlighting consumer preferences and purchasing patterns in the households studied.

References

  • Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson Education.
  • Ross, K. (2010). Elementary Statistics. Academic Press.
  • Johnson, R. A., & Wichern, D. W. (2014). Applied Multivariate Statistical Analysis. Pearson.
  • Swanson, B. (2015). Set Theory and Probability. Wiley Publications.
  • Wasserman, L. (2013). All of Statistics: A Concise Course in Statistical Inference. Springer.

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