Question

If the manager of a bottled water distributor wants to estimate, 95% confidence, the mean amount of water in a 1-gallon bottle to within ±0.006 gallons and also assumes that the standard deviation is 0.003 gallons, what sample size is needed?

If a light bulb manufacturing company wants to estimate, with 95% confidence, the mean life of compact fluorescent light bulbs to within ±250 hours and also assumes that the population standard deviation is 900 hours, how many compact fluorescent light bulbs need to be selected?

If the inspection division of a county weighs and measures department wants to estimate the mean amount of soft drink fill in 2-liter bottles to within ± 0.01 liter with 95% confidence and also assumes that the standard deviation is 0.08 liters, what sample size is needed?

An advertising executive wants to estimate the mean amount of time that consumers spend with digital media daily. From past studies, the standard deviation is estimated as 52 minutes. What sample size is needed if the executive wants to be 95% confident of being correct to within ±5 minutes?

Answer 1

We can use the formula for sample size for a population mean with a specified margin of error and confidence level:
```
n = (Z^2 * σ^2) / E^2
```
where:
- Z is the z-score corresponding to the desired confidence level (in this case, 1.96 for 95% confidence)
- σ is the population standard deviation
- E is the desired margin of error

Substituting the given values, we get:
```
n = (1.96^2 * 52^2) / 5^2
n ≈ 385.07
```

Rounding up, we get a required sample size of 386.

Therefore, the advertising executive should sample at least 386 individuals to estimate the mean time that consumers spend with digital media with a margin of error of ±5 minutes and 95% confidence level.

Answer 2

To calculate the required sample sizes for the given scenarios, we can use the formula:

n = (Z * σ / E)^2

where:
n = required sample size
Z = Z-value for the desired confidence level (for 95% confidence, Z ≈ 1.96)
σ = standard deviation
E = desired margin of error

Let's calculate the sample sizes for each scenario:

1. Bottled Water:
Z ≈ 1.96, σ = 0.003 gallons, E = 0.006 gallons
n = (1.96 * 0.003 / 0.006)^2 ≈ 384.16
Since we can't have a fraction of a sample, we round up to the nearest whole number. Therefore, a sample size of 385 bottles is needed.

2. Compact Fluorescent Light Bulbs:
Z ≈ 1.96, σ = 900 hours, E = 250 hours
n = (1.96 * 900 / 250)^2 ≈ 49.96
Again, rounding up to the nearest whole number, a sample size of 50 light bulbs is needed.

3. Soft Drink Fill:
Z ≈ 1.96, σ = 0.08 liters, E = 0.01 liters
n = (1.96 * 0.08 / 0.01)^2 ≈ 122.76
Rounding up, a sample size of 123 bottles is needed.

4. Digital Media Consumption:
Z ≈ 1.96, σ = 52 minutes, E = 5 minutes
n = (1.96 * 52 / 5)^2 ≈ 384.16
Rounding up, a sample size of 385 consumers is needed.

Please note that the sample sizes calculated here assume a simple random sampling method and certain assumptions about the population.