Final answer:
To ensure the estimate of the mean number of new toys children buy each year has an error of at most 0.14 at the 98% level of confidence, the minimum sample size required is 801, after rounding up the calculated sample size which is approximately 800.50721.
Step-by-step explanation:
To find the minimum sample size required to estimate the mean number of new toys children buy each year with a specified margin of error at a given level of confidence, we use the formula for the sample size of a mean:
n = (Z*(σ/E))^2
Where:
- n is the sample size.
- Z is the z-value corresponding to the desired level of confidence (in this case, 98%).
- σ is the population standard deviation (in this case, 1.7).
- E is the desired margin of error (in this case, 0.14).
Since the level of confidence is 98%, we find the z-value that corresponds to the upper 1% of the distribution because 100% - 98% = 2% and 2%/2 = 1% (as we consider both tails of the normal distribution).
The z-value from the z-tables for 98% confidence level (or upper 1%) is approximately 2.33. Now, we can calculate the sample size:
n = (2.33 * (1.7 / 0.14))^2
n = (2.33 * 12.143)^2
n = (28.2929)^2
n = 800.50721
Since we need a whole number for the sample size and we round up to ensure the margin of error is not exceeded, the minimum sample size required is 801.