a factory produces 150,000 lightbulbs per day. the manager of the factory estimates that fewer than 1,000 defictive bulbs are produced each day . in a random sample of 250 lightbulbs , there are 2 defictive bulbs . determine whether the managers estimate is likely to be accurate. explain.

Question

a factory produces 150,000 lightbulbs per day. the manager of the factory estimates that fewer than 1,000 defictive bulbs are produced each day . in a random sample of 250 lightbulbs , there are 2 defictive bulbs . determine whether the managers estimate is likely to be accurate. explain.

Answer 1

Set up the proportion 150,000 over x= 250 over two. Cross multiply 150000 and two to get 300000. Set up the equation 300,000=250x. Divide both sides with 250 to get the answer.

Answer 2

To determine whether the manager's estimate is likely to be accurate, we can use hypothesis testing.

First, let's define our null hypothesis (H0) and alternative hypothesis (H1):

H0: The manager's estimate is accurate, and the true proportion of defective bulbs produced is less than 0.0067 (1,000/150,000)
H1: The manager's estimate is not accurate, and the true proportion of defective bulbs produced is equal to or greater than 0.0067

Next, we use the binomial distribution formula to calculate the probability of getting 2 or more defective bulbs in a sample of 250, assuming the null hypothesis is true.

We use the calculation P(X ≥ 2), where X is the number of defective bulbs:

P(X ≥ 2) = 1 - [P(X = 0) + P(X = 1)]

To calculate P(X = 0), we use the binomial distribution formula:

P(X = 0) = C(250, 0) * (0.0067)^0 * (1 - 0.0067)^(250 - 0)

Where C(250, 0) is the number of combinations of choosing 0 defective bulbs out of 250.

Similarly, to calculate P(X = 1), we use the binomial distribution formula:

P(X = 1) = C(250, 1) * (0.0067)^1 * (1 - 0.0067)^(250 - 1)

Using a statistical calculator or software, we can evaluate these probabilities to obtain:

P(X ≥ 2) ≈ 0.048

Since the probability of getting 2 or more defective bulbs in a sample of 250, assuming the null hypothesis is true, is approximately 0.048, which is less than the commonly used significance level of 0.05, we fail to reject the null hypothesis.

Therefore, there is not enough evidence to suggest that the manager's estimate is inaccurate. The sample data does not provide strong evidence to support the claim that the true proportion of defective bulbs is equal to or greater than 0.0067.

Answer 3

% Estimated=(1000/150000) * 100%=0.67 %

% Defective = (2/250) * 100% = 0.8 %.

(0.8% /0.67%) * 100% = 119%.
119% - 100% = 19 %.

The actual % is about 19% higher than the estimated %.