Determine the area under the standard normal curve that lies to the left of \( ( \) a) \( Z=-0.68 \), (b) \( Z=-0.99 \), (c) \( Z=0.78 \), and (d) \( Z=0.77 \). (a) The area to the left of \( Z=-0.68 \) is 0.2483 . (Round to four decimal places as needed.) (b) The area to the left of \( Z=-0.99 \) is 0.1611 . (Round to four decimal places as needed.) (c) The area to the left of \( Z=0.78 \) is 0.7823 . (Round to four decimal places as needed.) (d) The area to the left of \( Z=0.77 \) is \( \square \). (Round to four decimal places as needed.)

To find the area to the left of \( Z=0.77 \), you can consult a standard normal distribution table or use a calculator equipped with statistical functions. When you look up \( Z=0.77 \), you will find that the area is approximately 0.7764. So, the area to the left of \( Z=0.77 \) is 0.7764 when rounded to four decimal places. In practical settings, understanding the area under the curve is vital for various applications, particularly in fields like psychology, finance, and quality control. For instance, in quality control, companies often use this information to determine whether a product meets specifications based on the normal distribution of measurement errors. Hence, knowing these areas helps in making data-driven decisions!