A study conducted by a TV station showed the number of televisions per household and the corresponding probabilities for ecach. Find the mean, variance, and standard deviation.
(in table)
Number of televisions
1 2 3 4
Probability P(X)
0.32 0.51 0.12 0.05
If you were talking a survey on the programs that were watched on television, how many program diaries would you send to ecach household in the survey?
To find the mean (expected value), variance, and standard deviation, we need to use the formulas:
Mean (expected value): μ = Σ(x * P(x))
Variance: σ^2 = Σ((x - μ)^2 * P(x))
Standard Deviation: σ = √(σ^2)
Using the given data:
Number of televisions: 1, 2, 3, 4
Probability (P(X)): 0.32, 0.51, 0.12, 0.05
First, we calculate the mean:
μ = (1 * 0.32) + (2 * 0.51) + (3 * 0.12) + (4 * 0.05)
= 0.32 + 1.02 + 0.36 + 0.2
= 1.9
Next, we calculate the variance:
σ^2 = (1 - 1.9)^2 * 0.32 + (2 - 1.9)^2 * 0.51 + (3 - 1.9)^2 * 0.12 + (4 - 1.9)^2 * 0.05
= 0.0192 + 0.0021 + 0.0096 + 0.001
= 0.0319
Finally, we calculate the standard deviation:
σ = √0.0319
≈ 0.1786
Now, to determine the number of program diaries to send to each household in the survey, we would need more information. The number of program diaries would depend on factors such as the desired sample size, target population, and survey objectives.
To find the mean, variance, and standard deviation for the given data, we will use the formulas:
Mean (Expected Value) (μ) = Σ(X * P(X))
Variance (σ^2) = Σ((X - μ)^2 * P(X))
Standard Deviation (σ) = √(Variance)
First, let's find the mean:
Mean (μ) = (1 * 0.32) + (2 * 0.51) + (3 * 0.12) + (4 * 0.05)
= 0.32 + 1.02 + 0.36 + 0.2
= 1.9
The mean number of televisions per household is 1.9.
Next, let's calculate the variance:
Variance (σ^2) = ((1 - 1.9)^2 * 0.32) + ((2 - 1.9)^2 * 0.51) + ((3 - 1.9)^2 * 0.12) + ((4 - 1.9)^2 * 0.05)
= (0.89^2 * 0.32) + (0.11^2 * 0.51) + (1.11^2 * 0.12) + (2.11^2 * 0.05)
= 0.2512 + 0.00603 + 0.1553 + 0.2315
= 0.64403
The variance is 0.64403.
Lastly, we can find the standard deviation:
Standard Deviation (σ) = √(Variance)
= √(0.64403)
= 0.8025
The standard deviation is approximately 0.8025.
Now, let's address the question about program diaries for the survey. The number of program diaries needed depends on the objective of the survey and the target population. However, a common approach is to send the program diaries to a representative sample of households rather than sending one to each household in the population.
To determine the sample size, you can use statistical methods such as sample size calculators, which consider factors like the desired level of confidence, margin of error, and population size. These tools help determine the appropriate sample size for accurately representing the population's preferences for TV programs.