A supplier has stated that the proportion of defective items in a very large batch of components is 5%. The customer has decided to check this statement by examining every item in a sample of 300 items selected at random from the batch. The sample was found to contain 21 defective items and 279 perfect items. Determine the lower limit, if a 95% confidence interval is set up for the proportion of all defective items.
To determine the lower limit of the confidence interval for the proportion of all defective items, we can use the following formula:
Lower limit = sample proportion - margin of error
To calculate the sample proportion, we divide the number of defective items in the sample by the total sample size:
Sample proportion = defective items / total sample size
= 21 / 300
= 0.07
The margin of error can be calculated using the following formula:
Margin of error = critical value * standard error
First, we need to determine the critical value for a 95% confidence interval. Since we are looking for the lower limit, we will use a one-tailed test. The critical value for a one-tailed 95% confidence interval is approximately 1.645.
Next, we need to calculate the standard error, which is the standard deviation of the sampling distribution. The formula for the standard error of a proportion is:
Standard error = sqrt[(sample proportion * (1 - sample proportion)) / total sample size]
Standard error = sqrt[(0.07 * (1 - 0.07)) / 300]
= sqrt[0.000203333]
= 0.014259
Finally, we can calculate the margin of error:
Margin of error = 1.645 * 0.014259
= 0.023463
Now we can calculate the lower limit:
Lower limit = sample proportion - margin of error
= 0.07 - 0.023463
= 0.046537
Therefore, the lower limit of the 95% confidence interval for the proportion of all defective items is approximately 0.0465 (or 4.65%).
To determine the lower limit of a 95% confidence interval for the proportion of all defective items, we can use the formula:
Lower Limit = Sample Proportion - Margin of Error
First, let's calculate the sample proportion:
Sample Proportion = Total Defective Items in Sample / Sample Size
= 21 / 300
= 0.07
Now, let's calculate the margin of error using the formula:
Margin of Error = Critical Value * Standard Error
Since the sample size is large (n > 30) and we are dealing with proportions, we can use the Z-distribution and find the critical value for a 95% confidence level. The critical value corresponds to a Z-score of 1.96.
Standard Error = sqrt( (Sample Proportion * (1 - Sample Proportion)) / Sample Size )
= sqrt( (0.07 * (1 - 0.07)) / 300 )
= 0.016
Margin of Error = 1.96 * 0.016
= 0.03136
Finally, let's calculate the lower limit:
Lower Limit = Sample Proportion - Margin of Error
= 0.07 - 0.03136
= 0.03864
Therefore, the lower limit of the 95% confidence interval for the proportion of all defective items is approximately 0.0386 (or 3.86%).