To find the number of light bulbs that are expected to last more than 1890 hours, we need to calculate the area under the normal distribution curve between 1890 and 1900 hours. This represents the probability that a randomly selected light bulb will last more than 1890 hours. We can then multiply this probability by the total number of light bulbs, 670, to find the number of light bulbs that are expected to last more than 1890 hours.
To calculate the area under the normal distribution curve, we first need to standardize the value 1890 hours. This means we need to convert it to a z-score, which represents the number of standard deviations above or below the mean a value is. To standardize a value, we use the following formula:
z = (x - mean) / standard deviation
In this case, we have x = 1890, mean = 1900, and standard deviation = 45, so the z-score is:
z = (1890 - 1900) / 45 = -0.22
We can now use this z-score to look up the corresponding area under the normal distribution curve using a table of the standard normal distribution. This table gives the area under the curve between the mean and a given z-score. Since our z-score is negative, we need to look up the area to the left of the mean. In this case, the area is 0.4162.
We can now multiply this probability by the total number of light bulbs, 670, to find the number of light bulbs that are expected to last more than 1890 hours. This is:
0.4162 * 670 = 278.29
Rounded to the nearest whole number, this is 278 light bulbs. So we would expect about 278 out of 670 freshly installed light bulbs in the new large building to last more than 1890 hours.
The answer is 278 light bulbs.