a pizza shop owner wishes to find the 95% confidence interval of the true mean cost of a large plain pizza. how large should the sample be if she wishes to be accurate to within $0.15? a previous study showed that the standard deviation of the price was $0.26
Cost of Pizzas A pizza shop owner wishes to find the 95% confidence interval of the true mean cost of a large plain pizza. How large should the sample be if she wishes to be accurate to within $0.15? A previous study showed that the standard deviation of the price was $0.26.
To find the sample size required to estimate the mean cost of a large plain pizza with a desired level of precision, you can use the formula for sample size calculation with a known standard deviation. Here are the steps to determine the required sample size:
Step 1: Determine the desired level of confidence.
The problem states that a 95% confidence interval is desired.
Step 2: Look up the Z-score corresponding to the desired level of confidence.
For a 95% confidence level, the corresponding Z-score is approximately 1.96. This value can be obtained from a standard normal distribution table or a statistical calculator.
Step 3: Determine the desired level of precision.
The problem states that the owner wishes to be accurate to within $0.15.
Step 4: Determine the standard deviation of the population.
The previous study showed that the standard deviation of the price was $0.26.
Step 5: Use the following formula to calculate the required sample size:
n = (Z^2 * σ^2) / E^2
Where:
n = sample size
Z = Z-score
σ = standard deviation
E = margin of error (desired level of precision)
Now, plug in the values:
n = (1.96^2 * 0.26^2) / 0.15^2
n ≈ (3.8416 * 0.0676) / 0.0225
n ≈ 0.2596 / 0.0225
n ≈ 11.52
Step 6: Round the sample size up to the nearest whole number.
Since we can't have fractions of a sample, round up to the nearest whole number to ensure an adequate sample size.
The owner should have a sample size of at least 12 to estimate the true mean cost of a large plain pizza with a 95% confidence level and be accurate to within $0.15.
To find the sample size required to construct a 95% confidence interval with a desired level of accuracy, we need to use the formula:
n = (Z * σ / E)^2
Where:
n = required sample size
Z = z-value for the desired confidence level (for a 95% confidence level, the z-value is approximately 1.96)
σ = standard deviation of the population (provided as $0.26)
E = desired margin of error (which is $0.15)
Plugging in the values, the formula becomes:
n = (1.96 * 0.26 / 0.15)^2
n ≈ 3.41^2
n ≈ 11.6081
Since a sample size must be a whole number, we round up the result to the next whole number:
n ≈ 12
Therefore, the pizza shop owner should have a sample size of at least 12 to be accurate within $0.15 with a 95% confidence level.
Use a formula to find sample size.
Here is one:
n = [(z-value * sd)/E]^2
...where n = sample size, z-value = 1.96 for 95% confidence, sd = 0.26, E = 0.15, ^2 means squared, and * means to multiply.
Plug the values into the formula and finish the calculation. Round your answer to the next highest whole number.