1. The new Twinkle bulb has a standard deviation σ = 34 hours. A random sample of 77 light bulbs is selected from inventory. The sample mean was found to be x= 492 hours.

Question: Find the margin of error E for a 90% confidence interval.

Round your answer to the nearest hundredths.

Answer:
n = 77, 492, σ = 34, zc = 1.645
1.645 x 34/√77 = 6.37

Question: Construct a 90% confidence interval for the mean life, µ of all Twinkle
bulbs.

Answer:
n = 77, 492, σ = 34, zc = 1.645
492 – 6.37 = 485.63
492 + 6.37 = 498.37

485.63 < µ < 498.37

To find the margin of error, we can use the formula:

E = zc * (σ / sqrt(n))

Given that σ = 34, n = 77, and zc for a 90% confidence level is 1.645, we can substitute these values into the formula:

E = 1.645 * (34 / sqrt(77))

Calculating this expression:

E ≈ 1.645 * (34 / 8.77496438739)
E ≈ 1.645 * 3.87526048454
E ≈ 6.37

Therefore, the margin of error (E) for a 90% confidence interval is approximately 6.37.

Now, to construct a 90% confidence interval for the mean life (µ) of all Twinkle bulbs:

Using the sample mean (x) of 492 from the 77 light bulbs, and the margin of error (E) of 6.37, we can find the lower and upper limits of the confidence interval.

Lower limit = x - E
Lower limit = 492 - 6.37
Lower limit ≈ 485.63

Upper limit = x + E
Upper limit = 492 + 6.37
Upper limit ≈ 498.37

Therefore, the 90% confidence interval for the mean life (µ) of all Twinkle bulbs is approximately 485.63 < µ < 498.37.

To find the margin of error, you need to use the formula:

E = zc * (σ / √n)

In this case, n is the sample size, σ is the standard deviation, and zc is the z-score corresponding to the desired confidence level (in this case, 90%).

Given that n = 77, σ = 34, and zc = 1.645, you can simply substitute these values into the formula:

E = 1.645 * (34 / √77)

Calculate the result:

E ≈ 6.37

Therefore, the margin of error for a 90% confidence interval is approximately 6.37.

To construct a 90% confidence interval for the mean life µ of all Twinkle bulbs, you need to use the sample mean (x) and the calculated margin of error (E).

Take the sample mean (x): 492

Then subtract the margin of error (E) from it:

Lower bound = x - E = 492 - 6.37 ≈ 485.63

Next, add the margin of error (E) to the sample mean (x):

Upper bound = x + E = 492 + 6.37 ≈ 498.37

Therefore, the 90% confidence interval for the mean life µ of all Twinkle bulbs is approximately 485.63 < µ < 498.37.