A 4 to 7 month old baby sleeps an average of 14 hours per day. At this age, babies sleep patterns follow normal distribution with the standard deviation of 1.6
a) is it unusual for a baby to sleep under 12 hours aday?
b) In a random sample of 10 babies of this age,find the probability that the mean number of hours of sleep is below 13 hours.
a) Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
b) Z = (score-mean)/SEm
SEm = SD/√n
Use same table.
To determine whether it is unusual for a baby to sleep under 12 hours a day, we need to calculate the z-score, which measures the number of standard deviations a particular value is away from the mean.
a) To find the z-score for a baby sleeping under 12 hours, we use the formula: z = (x - μ) / σ, where x is the value we want to find the z-score for, μ is the mean, and σ is the standard deviation.
First, let's calculate the z-score for 12 hours of sleep:
z = (12 - 14) / 1.6
z ≈ -1.25
Next, we evaluate the z-score using a z-table or a statistical software. The z-table provides the proportion of values below a given z-score. A z-score of -1.25 corresponds to approximately 0.1056, or 10.56% of values below it.
Since this percentage is relatively low, it can be considered unusual for a baby of this age to sleep under 12 hours a day.
b) To find the probability that the mean number of hours of sleep is below 13 hours in a random sample of 10 babies, we need to calculate the sampling distribution of the mean and find the z-score for 13 hours.
The sampling distribution of the mean follows a normal distribution as the sample size is large (assuming the sample is random and independent).
The mean of the sampling distribution of the mean is the same as the population mean, which is 14 hours.
The standard deviation of the sampling distribution of the mean, also known as the standard error, can be calculated using the formula: σ / √n, where σ is the standard deviation of the population and n is the sample size.
In this case, the standard deviation of the population is 1.6, and the sample size is 10. Therefore, the standard error is 1.6 / √10 ≈ 0.507.
Now, we can calculate the z-score for 13 hours:
z = (13 - 14) / 0.507
z ≈ -1.975
Using the z-table or statistical software, we can find that the proportion below a z-score of -1.975 is approximately 0.025, or 2.5%.
Therefore, the probability that the mean number of hours of sleep is below 13 hours in a random sample of 10 babies is approximately 0.025 or 2.5%.