Step 1 :The P-value is the probability that we would observe a test statistic as extreme as the one we calculated, assuming the null hypothesis is true. In a left-tailed test, we are interested in the probability of observing a test statistic as extreme or more extreme in the negative direction. However, our test statistic is positive. This means that our P-value is going to be very high, specifically it's going to be greater than 0.5.
Step 2 :To calculate the exact P-value, we need to use the cumulative distribution function (CDF) for a standard normal distribution. The CDF gives us the probability that a random variable drawn from the given distribution is less than or equal to a certain value.
Step 3 :However, since our test statistic is positive and we are conducting a left-tailed test, we need to find the probability of observing a test statistic as extreme or more extreme in the positive direction. This is simply 1 minus the CDF at our test statistic.
Step 4 :Given that the test statistic Z = 1.62, we find that the P-value = 0.053.
Step 5 :Final Answer: The P-value for a left-tailed hypothesis test with a test statistic of \(Z=1.62\) is \(\boxed{0.053}\).