5.7 Sampling Distributions for Sample Means

Cards (42)

  • The sampling distribution of sample means approaches a normal distribution as the sample size increases, even if the population distribution is not normal.

    True
  • The standard deviation of the sampling distribution of sample means is \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}</latex>.
  • The Central Limit Theorem for sample means applies regardless of the population's original distribution.

    True
  • The mean of the sampling distribution of sample means is equal to the population
  • Match the property with its corresponding formula:
    Mean of sample means ↔️ μxˉ=\mu_{\bar{x}} =μ \mu
    Standard deviation of sample means ↔️ σxˉ=\sigma_{\bar{x}} =σn \frac{\sigma}{\sqrt{n}}
  • What is the sampling distribution of sample means defined as?
    All possible sample means
  • Match the property with its description for the sampling distribution of sample means:
    Mean ↔️ Equal to the population mean
    Standard Deviation ↔️ Population standard deviation divided by the square root of the sample size
  • Under what condition does the sampling distribution of sample means approximate a normal distribution according to the CLT?
    n≥30n \geq 30
  • What is the formula for the standard deviation of the sampling distribution of sample means?
    σxˉ=\sigma_{\bar{x}} =σn \frac{\sigma}{\sqrt{n}}
  • The mean of the sampling distribution of sample means is equal to the population mean.

    True
  • The Central Limit Theorem is crucial for making valid statistical inferences about population means.
    True
  • μ\mu represents the population mean in statistical formulas.

    True
  • Steps for making statistical inferences using the sampling distribution
    1️⃣ Collect sample data
    2️⃣ Calculate the sample mean
    3️⃣ Determine the properties of the sampling distribution
    4️⃣ Construct a confidence interval or conduct a hypothesis test
  • What type of inference is used to estimate the population mean using a confidence interval?
    Estimation
  • What is the sampling distribution of sample means?
    All possible sample means
  • Why is the sampling distribution of sample means important?
    Statistical inference foundation
  • The Central Limit Theorem (CLT) states that the sampling distribution of sample means becomes approximately normal when the sample size is sufficiently large.
    True
  • What is the mean of the sampling distribution of sample means according to the CLT?
    μxˉ=\mu_{\bar{x}} =μ \mu
  • The standard deviation of the sampling distribution of sample means is always equal to the population standard deviation.
    False
  • If a population has a mean of 50 and a standard deviation of 10, the mean of sample means with a sample size of 25 is 50
  • The mean of the sampling distribution of sample means is equal to the population
  • The Central Limit Theorem ensures that the sampling distribution of sample means becomes approximately normal with a large sample size, regardless of the population's shape.
    True
  • The standard deviation of the sampling distribution of sample means is calculated as \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}</latex>
  • The sampling distribution of sample means is approximately normal if the sample size is 30 or more.

    True
  • The standard deviation of the sampling distribution of sample means decreases as the sample
  • The mean of the sampling distribution of sample means is denoted as \mu_{\bar{x}}</latex>
  • If a population has a mean of 50 and a standard deviation of 10, the mean of the sampling distribution of sample means with a sample size of 25 is 50
  • Statistical inference by estimation involves determining a likely range for the population mean
  • A hypothesis test can determine if there is enough evidence to reject a claim about the population mean.

    True
  • The standard deviation of the sampling distribution of sample means is the population standard deviation divided by the square root of the sample size.
  • What is the mean of the sampling distribution of sample means equal to?
    Population mean
  • What is generally considered a sufficiently large sample size for the Central Limit Theorem to apply?
    30 or more
  • What does the Central Limit Theorem (CLT) for sample means state about the sampling distribution of sample means?
    Approximates a normal distribution
  • What is the general condition for the sample size to apply the CLT for sample means?
    n≥30n \geq 30
  • The standard deviation of sample means with a sample size of 25 from a population with a standard deviation of 10 is 2.

    True
  • How does the standard deviation of the sampling distribution of sample means relate to the population standard deviation and sample size?
    σxˉ=\sigma_{\bar{x}} =σn \frac{\sigma}{\sqrt{n}}
  • The mean of the sampling distribution of sample means is equal to the population
  • The Central Limit Theorem states that the sampling distribution of the sample means approximates a normal
  • What are the conditions for applying the Central Limit Theorem?
    Random samples, n≥30n \geq 30
  • What is the formula for the standard deviation of the sampling distribution of sample means?
    σxˉ=\sigma_{\bar{x}} =σn \frac{\sigma}{\sqrt{n}}